Hey guys, it's Nate, aka The Foot Accountant. Welcome back to the channel. Another promo Friday is upon us. Does this World Cup Stories promo have the hype to bring some hype back to FIFA today? As the past couple of days in this game have just not been that extra special, right? They've just been okay with a road to the World Cup cards and the World Cup icons. While we're getting new World Cup icons today and brand new cards and packs as the previous promo cards will be going out of packs, I want to take a look at what is to come with this promo today, what we expect to see, and then also regarding Ones to Watch and Path to Glory upgrades today, late tonight, not around content time. If EA do it like they did earlier in the week, we could have our second round of upgrades to these live cards and of course to the Ones to Watches like Dest, like Tyler Adams, and some of those other ones like Rudeger and Schlotterbeck that are now deserving of some upgrades. So I want to take a look at that and also talk about the market because since the game has not been that hype, where the content has not been super duper hyped the past couple of days, is today going to be a day that kind of rises that hype back up with the new promo? Or will it just be a day of supply and lightning rounds and maybe not a whole lot going on? We're going to take a look at that and more in today's video. If you're excited for it, hit the thumbs up, subscribe if you're new. Let's start by diving into the loading screen that we had just really quick. And again, we don't know what this promo is exactly about, but we're starting to get a good guess because of the cards that are shown on the loading screen, they match up to the leaks except for this card in the middle, 91 passing and Portuguese. It would make sense with the way that these cards are looking via the leaks, that that would be Bruno Fernandez. Some people are saying João Felix, but that'd be a huge passing upgrade. And if it makes sense to you like it does to me, a lot of the leaks that we have right now are for cards and players that have performed well at the World Cup. Stories of the World Cup so far this year as the tournament has been going on, right? Richard Wilson, arguably goal of the tournament. This card, if these are the exact stats, are not very good. It's just a small plus one over his wants to watch card if you have that one. But of course, being recognized with a card in the game for his insane goal that he had during the tournament. This is our USA card as an objective today, right? This is our objective according to the leaks. Mr. Tim Wea, that's a very pacy card fits the meta and my US boys would love to get that done. So there's our objective card there. It's a nice one right there. Moving on, we've got a Jacques card that to be completely honest, just looks like the Paulina SBC, even worse than that. So that's going to be a pretty cheap card. Probably not that hype unless you're a Arsenal fan or a Swiss fan for that one right there. We have an Alfonso Davies left wing. Some of these leaks you guys may have already have seen, but I want to scroll through them pretty quick. Hopefully he has left back available to him as well. But you know, a moments for our stories card for Alfonso Davies for scoring the first Canada goal at the World Cup, right? That's deserved for him there as well, even though they're now out of the tournament. But that's kind of how this promo seems to be right. A lot of these players, as we've seen the leaks, is just a representation of players that have performed somewhat well at the World Cup so far, whether it's been scoring goals or holding clean sheets for some of the defenders. Asensio scored some goals. La Porte writes, Bane has been doing very, very well, apart from yesterday's game against Japan. And then of course, the early on leaks, we have Sane, you guys have seen these. We have Rashford and you guys have seen the first one, of course, Van Dyke. So that's one of the cards in the loading screen. We believe it's Van Dyke, Bruno Fernandez and Asensio. So we're a little unsure. There's probably going to be more leaks today talking about what cards are going to be out. We also don't know what SBC will be today. So it's kind of nice to have a little bit of a surprise heading into a promo Friday. We know the objective is way up. We don't know about our SBC yet. And we will probably see some more leaks today about different cards that are going to be in packs as well. So that's what is happening today. Now, we also know we're going to get new icons in packs. So for you guys that we're looking to pack Figo or Jogbo or trying to, you know, maybe even to buy a World Cup icon from Team 1, these cards are going out of packs. Like this is Ambrata, who was in a mini-release. I traded with one this evening. I bought it for 40, sold it for 480. He is up a little bit. A lot of these World Cup icons, just because they're so rare and they're pretty hyped up, I think they will rise a little bit out of packs. Probably not anything super-duper crazy, but you know, some slight rises as the team two of the World Cup icons is dropping today doesn't seem to be that incredibly hyped. I think Team 1 is going to end up being better. So that's just something to watch out for. But we'll have new World Cup icons in packs today as well as a part of the content. And that's probably going to be it. Right? Again, what really matters today in terms of is it going to be a hype day on the game is the SBC content, right? We know the objective. We don't know the SBCs. Will EA surprise us again this week with like a big upgrade pack SBC on a Friday? In my opinion, they probably will not. And that was a thing that we saw last week with Black Friday, right? We had like the 87 plus icon and the 86 plus foot hero upgrade on the weekend. I think one of those is on a Saturday, but I don't think we're going to have a big upgrade SBC like this re-release today. We will still have the 86 plus hero in the game today. So there will still be people maybe trying to go in to complete that one. But a lot of your upgrade packs, of course, the year interview player pick the 87 max icon that goes away or 87 plus icon goes away today at content time. So those are going to be going away. We'll talk about that in relation to fodder in a second. But the way that it seems and with the way that content has been going on this game, is it going to be a really hype day today? What? Where's the hype? Right? And first of all, you think about this. You say, is there going to be panic selling for most of the market? Like, let's say you have a Gold Christian Ronaldo and your team, you're like, Nate, I bought this Ronaldo guy last week at like 80 K and now he's 95. Do I need to sell this card today? To be honest, I don't think you need to panic sell your team today. There's really, as I look at it, not much content is dropping today apart from some cool new cards to try out. They would really make the market crash. I'm not expecting a market crash today. The only thing I'm a little worried about is the supply. Once again, like we saw last week, we had lightning rounds and usually Black Friday is the first weekend of the year where we have a lot of lightning rounds in FIFA. And then every single weekend after that, we continue to get lightning rounds. So I do believe that we'll see 50 K, 100 K, 125 and maybe some of those star player pack lightning rounds throughout the day today on Friday and into the weekend with this new promo being out. So just watch out and expect those. Now I'm also pretty curious about the pack weight today because as you can see, road to the FIFA World Cup players, 8.6% out of a 100,000 coin pack, 4.3% out of a 50 K pack, which I believe this has been updated because if I remember correctly last week on Black Friday out of a 50 K pack, the road to the World Cup players, it was only like a 1% chance and I think EA had the pack weight turned way down last Friday because so many packs were going to be open with all the lightning rounds and everything in the store at that time. So I think they may have turned the pack weight down. That makes me think that maybe some of the World Cup stories cars that drop on the game today will be supplied a bit more and maybe we'll end up being a little bit cheaper because of a higher pack weight. So we'll just have to see what it's like when they drop on the market today. Also, it depends. Will they drop another store pack? In my opinion, I think they will because I really do feel like EA is trying to work in some more tokens in the store. We only have had to write the two foundations packs from last weekend. So watch out for another store pack today. They could literally just re-release one of the same ones that they had, I guess, even though one of them was called Black Friday. Maybe we'll have a store pack today to keep an eye out for that as well. So again, in general, when we talk about panic selling for the market today, I wouldn't expect to see a lot. I mean, I think your path to glory cards if you've invested in those, you're looking for the upgrades coming later tonight later, maybe into tomorrow on Saturday in the early morning hours. I really wouldn't be panic selling any of these. Of course, for the clubs, for the, for the natures that play today, maybe you take a look at selling in the hype just in case you want to take the safe profit there. If there's a situation today, maybe if it's like you've got Malinkovic Savage and they're on the verge, you know, they're right now not in the advancing, but you know, with a win, they could advance type of thing. So maybe you sell them to the hype with a few of those cards or whatever. But, you know, I would just not expect a lot of panic selling today just because I don't think a lot of people are scared about what is to come content wise today. It's been a pretty quiet week and it might just be one of those promo Fridays where there's not that much hype unless EA do something special. I think again, like I said, the place where they would do that is in SBCs. We just have to wait and see what kind of content they drop today and EA surprise us, man. Drop something sick. We would love that. Now with all the supply coming today, I do want to talk about SBC fodder because this is a part of the market that needs to drop a lot and I think it's going to start dropping today.  We had supply yesterday. 85s are still pretty high, but they're going to come down in my opinion. 86s are going to come down. Think about what we talked about, right? That 87 max icon SBC is going away, right? That SBC is going to expire today. You still have the hero SBC that is out there, but I think most people at this time have probably done at least that SBC once and the demand for that SBC is probably continuing to dwindle over the next day or two. I still think that this SBC will help the fodder market a little bit, but with all the supply and with a lot of these other upgrade packs like the player picks and stuff in here that people use to grind and craft SBCs like this one going away, I think that, you know, you will see the demand for those fodder cards drop. So especially with this SBC going away and this year in review going away, I think you're going to see some of the fodder market drop off. But I would say watch out for the supply today because this SBC specifically requires those 85s. And I think that's why the 85s are maintaining a pretty high price. People are gambling on this pack because it's not that expensive. It's just cheap enough that they can do it. If they get some fodder cards from upgrade packs or weakening rewards or whatever, this is still available and I'm sure some people haven't finished it yet. So watch your 85s today because I think besides the 85s, most of the fodder market should probably keep dropping though. I think your 84s are probably going to drop a bit more. Your 86s will go down a bit and I think your high tier fodder too will probably trickle off today because a lot of that high tier fodder was needed in the 87 plus icon pack with an 87 squad, with an 85 squad in there. And some other people doing icon SBCs, maybe doing messy, messy expires today. He had a high rated squad as a part of his SBC as well. So I think your high tier fodder is going to start to trickle down today with the Lightning round supply and all the packs that we will have there. Now it could be a good look this weekend later on. So just keep an eye out for those prices dropping today on the game. That's kind of what I would expect to happen with those. Now, arguably more hype than the promo starting today for some people is the path to glory cards and the ones to watch cards that should be getting upgraded today. Now we say should, right? We're saying should because according to how EA did things earlier in this week, they upgraded these path to glory cards like Bruno Gamerich, the French guys like in Kunku or Griezmann. These guys got upgraded after their matchday two because they clinched the spot in the knockout rounds, right? And of course, a lot of these cards have gone up in price a ton. Griezmann is up like a hundred K since he got upgraded. I think that your second batch of upgrades is coming tonight, Friday night into Saturday, midnight UK time. That is the exact time that EA tweeted it out earlier in the week when we had upgrades to some of those cards like the Retrylaton wants to watch the Antony wants to watch. Now tonight there's going to be a ton of upgrades. If that does end up being the case, you're going to look at guys like Dest and Adams that are going to get upgraded. Of course, your PTG cards, you know, some of the most popular guys like that are advancing from the games earlier on this week, Christian Pulisic, Ito, Bufal, Brozovic, Carval Haldes is going to get upgraded. Greelish, Kyle Walker, Kula Bali, Millick, those are going to be some of the PTG cards the young. There's going to be a lot of cards that will be upgraded late today. Hopefully if they follow that trend again, it's not a 100% guarantee, but it seems very, very likely that they will do that since of how they did it early in the week. But there are some nice upgrades coming today or coming very soon with these cards. So I would again, if the content is not that exciting or not that hype today, probably after the upgrades, these cards are just going to kind of trickle back up in price a little bit. Now I would say that also some of these cards could end up going down in price because this will be a time where especially for a guy like maybe Ito who's not expected to progress in the tournament that much further or maybe Bufal for some of these guys that were kind of like surprises, if you will. Not that they're playing poorly, of course, they deserve to be in the knockouts after their performances, but surprise is in the fact that they're not expected to go too much further in the tournament. These sorts of cards may drop off a little bit as they're probably expected to lose a little bit later on as the knockouts start tomorrow on Saturday. So just kind of watch out for that, but especially with some of these upgrades, like I'm super excited for the Pulisic card. I've got one of these. I've sold two now at 97K for Lazies. That's a nice undercut there on Pulisic, but you know, I just think that these cards are going to maintain their prices decently well just with how the way the market is right now and just being kind of quiet. So watch out for those. There should be plenty of movements on the market related to that. Now let's talk trading today too, right? Because if the market's going to be a bit quiet today, what kind of stuff would you want to be looking out for if you're trying to make coins? Well, I'd honestly say that if you're trying to make coins, what we always do is trading with the brand new promo cards is always high because people want to try these cards out, right? Even though we say this promo might be a little bit dead, you got a promo card for Rashford, right? It's a big upgrade for him. You got a Sony card with maybe a week foot upgrade. Who knows? You've got a La Porte card, left footed center back with 80 pace making this card usable. There will be plenty of promo cards in this team today or the World Cup icons that will be very, very tradable with. Now, I don't know what the trends are going to look like. I can't exactly say far ahead of time when the cheapest point is going to be for a lot of these World Cup stories cards inside of the first couple hours. I doubt it would be the first hour. I don't think there's a ton of saved up packs. So just kind of watch these cards as they move on the market because people will be excited to try the new cards that drop on fever like they are every week. Now, I will say for the World Cup icons, if you'd like to trade with icons, this last Friday was a fantastic day. I remember specifically the petite card rose like 150K after reaching a very, very low point on that first night. Dragba last Friday night was like 700K or 670. He went all the way above like 800K or something like that on that first night. So really, really, really watch the World Cup icons because there's a lot of World Cup icons that are dropping today that are kind of probably going to be in that like 300,000 coins, like a million coins threshold. So watch in that range. If you want to try one of them out, just be monitoring their prices because we will get supplied today and there will be fluctuation on them. But I would also say there should be some really good opportunities to trade with them as well. And when it comes down to the promo cards that we have right now, the Road to the World Cup, these cards, you know, I think that we'll see a little bit of rise out of pack on some packs on some of these. Not saying they're all going to absolutely explode, but I would say keep an eye on some of your most popular ones like the Marquinhos that we looked at here. I flipped. I actually got a really unlucky snipe on a Marquinhos earlier today. I bought a Marquinhos at 500,000 coins. Somebody must have packed him from Marquinhos matchups and just listed him up. And instead of a six, this at a five or still, that's a very, very cheap price. But anyways, I got a good snipe on this guy yesterday and he was a good profit for me. But I think that this guy is so rare. I think that the Sonaldo card is so rare that there will be great opportunities to be trading with some of those cards in the market today. Watch your icons. Watch your path to glories. Anything that is meta on the game today should do pretty well. And watch, you know, especially some of these path to glories. I've been, you know, watching this Gretzka tonight. You know, Gretzka and the Germany cards, of course, after one of the most insane days of football I think I've ever seen. And what a day. 449. I will take that. Thank you, Gretzka. That's a 50K undercut. And he was like 520K earlier. I'm going to try to quick flip this Gretzka card now. That's just what we're going to do. I don't know how that card was just sitting there for five minutes. 50K, 45K under the cheap is buy it now. But this Gretzka card for me into today on Saturday, I'm watching it, right? Because there's a lot of panic selling post the game. And there is a lot of supply. A lot of people are listing and trying to get out. The one thing I don't like and probably why I'm going to try to make a quick flip out of this is all of the overnight listings.  All right. If you're going to buy a card to flip, don't be the guy that lists up a bunch of them at 550 overnight for a couple of hours. Like, there's a whole bunch of these listed with zero games played current club. That is not how you want to list your cards if you're going for like an overnight flip. Legit. If you bought like more than one or two for an overnight flip. If you're on a huge, huge budget, then honestly, what I would say to do is list one or two and then wait and see what the price is.  Wait to see what the price is after you wake up if you're listening overnight and then go to sell the card as you monitor the price, like one at a time, right? It's just not a place on the market where you're going to be mass listing cards. So I've got a Gretzka. I did pick up a Keraska or two at by 51,000 coins. I did also pick up a Serge Nabri at 35K. I know he's literally minimum price. He cannot go any lower. The reason I picked that up is because I think this card in a week's time or two weeks time as long as EA don't update the price range today or tomorrow while he's being panic sold still. I do think that he'll end up being around like a 45,000 coin card because again, Byron links, you know, German links. This is just a card that a lot of people use because it's Byron. And again, these types of links are so important with the chemistry that the way that it is this year in FIFA. So I would absolutely keep an eye out on this Nabri card. I picked one up for Discard because I think there's not a lot of risk there even though he is minimum price. Now for Lukaku I don't like Lukaku as much. I like Nabri more 412,000 games played for Lukaku. How many for Nabri? Nabri's got 519,000 games played. So more games played for him also bodes well for his price to rise back up a little bit just because of the hype of that card and the usability of it. How many is Pulisic have 141,000? Wow, that's crazy. I guess Pulisic was you know, in the second team. So that would hurt his time as well not having as much time in the game. But you kind of see what I'm going with there. I'm trying to trade with the cards that are the most used that are the most popular. If again, if you're trying to trade for today, watch the rare cards. Icon should be good today.   Let's see what SBCs we get. Open a couple packs. See if we hit anything. Maybe do an objective or two. Oh, last thing. We should. We should get some more tokens today. I think we might get four more tokens. That's been the Friday trend. They dropped the showcases on the Fridays where they dropped like the rivals squad battle tokens. So right now we're on 40 tokens in total. They dropped four more today. We'd be at 44 potentially a store token. Maybe maybe take us to like 45. So if you're on the token grind right now, maybe a couple more tokens today, I would watch out for that being the store as well. So we'll just see, right? It's kind of nice rolling into a promo Friday where it's like, all right, Esports, let's see what you got, right? If you're going to drop something sick, then show us. And if EA, if you would like to fix the menus here, that would be fantastic. I'm getting really tired. It's having to search every single time or going and searching in the squad menu screen or cards instead of searching on the market here because every time you back out, it of course, removes your search. So that is very annoying. As you can see, as just did right there. So hopefully EA fixed that very soon. And yeah, guys, that's going to be the video for today. Hopefully it's a great promo Friday. I'll be live streaming the content drop. We'll be checking out the new cards, checking out the game and see what EA has a store for us today on promo Friday. So if you're excited for the promo today, hit a thumbs up on the video. Comment down below if you have any questions and subscribe if you're new. It's been Nate's foot account. I'll catch you guys later. Peace out.


everyone to your third session. Today we're going to do question and answers based on study unit 4. And I've shared with you the weekly session plans. You must please follow that because it's paced in a way that it should help you understand and unpack the the material and be able to follow so that you don't fall behind to help you submit your assignments on time as well. And I've I've I've pasted them in a way that it should help us that we finish with the content a week before this assignment closes so that then it gives you more time if you still want to work on your assignments as well. You can still work on after our last session to make sure that you submit on time. With assignment one remember you will get your third submission right at the end of a week before the closing date. They will give you a a third submission and this is based on mainly because we started eTutorials late but this will not be for all the assignment. It might be but it might not be for all the assignment. It's only for assignment one and that is the reason why I'm saying and because we realize that also many people submitted their assignment one submission one and two one after the other. So we're allowing you to have a third submission because now we relate the message to you to say pace yourself. Don't do submission one and two at the same time. Allow time in between for the both of those two submissions. So now you should be able to do that. Okay so let's continue with today's session which is question and answers for study unit four which is basic probabilities. I've also asked you to go through the content because I post the content before a recording of the content before so that you are able to go through the two hour session or the one hour session of content so that when we come and do the question and answer session we concentrate on how to answer the question to relate into that content that you went through on your own. Right so I'm going to do a quick summary hoping that by half past that should be done with the quick summary and then we will start with the question and answer. If there is something that you don't understand please stop me. Don't raise your hand because I can't see my chat function. Please unmute and let me know that you have a question to ask before I continue and carry on. Don't just write on the chat. If there is a question on the chat somebody just let me do that as well. Okay so when we talk about probabilities we're talking about chances we're talking about the likelihood of something happening so a probability is a chance that an uncertain event will occur and that can always be between the values of zero and one. An impossible event is an event that cannot happen or it does not have any chance of happening and that will always have a probability of zero. An uncertain event is an event that we know for sure that it will happen and it will always have a probability of one. In terms of assessing probabilities as well there are three approaches and usually we use the a priori and empirical probabilities when we work with basic probabilities we don't use the subjective probabilities. You can go and study what all three of them do but a priori is based on the prior knowledge of the process. So if for example creating an event an event is like tossing your coin and when a coin lands on a head I should be able to calculate the probability of a coin landing on a head and that is based on the fact that I know that a coin has a head and a tail and it's a 50-50 chance that it will land on a head and that we calculated by using your x divided by t where x is the outcome that you are getting from that event that you created and your t is your sample space or your grand total or your total number of outcomes all of them combined that will give you your t and also with empirical probabilities it also uses your prior knowledge but that will also you will also have calculated the same way and that will be the number of ways the event can happen which are your outcomes divided by the grand total or the sample space or the number the total number of outcomes that you have and a subjective one is based on a researcher's own opinion so you can have your own opinion in terms of that it might be 70% of a chance that the sun will come out or it's gonna be raining and all that and those who are subjective to the researchers as well. I've already alluded to an event and we know that an event happens when you create an event it's like when you tossing a coin or you pulling a cat or rolling a die or the sun coming up that is an event an event has possible outcomes that can happen so when I roll a die a die has six sides so it can either land on a one or two or three or four five or six those are outcomes so when I do one thing when I do one event it is what we call a simple event rolling a die is a simple event when I do two things rolling a die and tossing a coin I am creating two events and those are called joint events an event that happens and they've got two or more characteristics that will okay and when I roll a die and I toss a coin and if it lands on a head and a die lands on a six I'm creating two joint events for that event that I am creating and a compliment is an event that does not include the others for example with a coin it has two sides a head it's a compliment of a tail a tail is a compliment of a head in case of a die one is a compliment of all the other sides of a die a six is a compliment of all other sides of a die and that's how you will read your compliment event so if I say I roll a die and a die lands on a one what is a compliment of a one it will be all the other events a die landing on a two three four five and a six those are a compliment of a one a sample space is a collection of all events or all outcomes so when you create an event an event is created from this one thing that has outcomes and those outcomes if I combine all of them they create what we call a sample space so a sample space we use that to create an event and within that event it will come out with an outcome like a die it's a sample space because all the sides of a die creates a sample space it creates a grand total the total of all sides and similar to a 52 cut of a cut a deck of a cut it's a sample space if I draw a cut I'm creating an event and the outcome can either be an ace of spade an ace of a diamond and an ace of a hard or a club things like that those are my outcomes okay we can visualize probabilities and events in terms of different things we can use a vain diagram and a vign diagram we can demonstrate simple event like the green cycle is a simple event of a if a represent aces so those will be the cuts that has an ace on it and be representing the cuts that are red and that will be that so we know that in a sample space which is a collection of all the cuts there are cuts that are red there are cuts that are black there are cut there is a cut that is an ace and an ace can be red or black and it's black when it's a club or spade and it's red when it is a heart or a diamond so in between where they both share because let a representing aces and be representing red cut therefore the ace of a diamond and an ace of a heart they are joined event or joint outcomes from ace and red and that creates a joint event as well and that we're going to discuss the joint event and the union events which are your A or B events we will get to that just now we can also visualize events using a contingency table and this is the most easy table to use if you are able to create this table you should be able to represent your event or your probabilities with this table and be able to answer questions and you will see that the majority of questions are based on a contingency table and if there is no contingency table we are going to create one to enable us to answer some of the questions and you can also create your or visualize your events based on a decision tree which also gives you the outcomes as well I'm not going to explain more about this because in that video that I'm explaining I have told you about I said you can go and listen to that or watch that it explains the type of events and what we see in those table in detail simple event and joint probabilities we calculate them by using the outcome satisfying the event divided by the grand total for a simple event so if I want to calculate the simple event like probability of a king I will use outcome satisfying a king divided by the sample space which will be the total of the deck which is 52 and because there are four kings in the deck so there will be four divided by 52 and that will give you your probability a joint event will be if I want to draw a cut that is a king and a spade so I'm creating two events at that point and that will be the number of outcomes satisfying that event which is a king and a spade there is only one cut that is a king and a spade and divide by there grand total or the sample space which will be 52 of them so it will be one divided by 52 when I use the joint event and that is how you will calculate the probabilities always probabilities it's going to be the probability of an event let's use a event a will be the number satisfying divided by the sample space even if it's a joint event event a and b you will realize that yeah I'm using and you can use and and you can use intersect as well they may one and the same thing and that will be observation satisfying that event the joint event divided by the grand total we're going to use that to calculate probabilities there are also what we call mutually exclusive events and those are events that will not happen or cannot happen at the same time and usually this are joint events so the probability of finding the mutually exclusive events event a and b if they are mutually exclusive the probability will be equals to zero because that is an impossible event that will happen we also have what we call collectively exhaustive event which means all events needs to satisfy the sample space so if I have event a which is representing aces and be representing black cards and see representing diamonds and be representing when also hearts event a b c and d will represent in the collectively exhaustive events because all of them make up the sample space but they are not mutually exclusive and ace may be also of a hat so a event a and g will not be mutually extra mutually oh they will not be mutually exclusive because we can have a cat that is an ace and a cat that is an ace with a hat and even b c and d are collectively exhaustive and are also mutually exclusive because a cat cannot be a hat and a black or it can also not be a diamond cat okay this is how you represent joint events and simple events so inside the contingency table those will be your joint event and you can calculate your probability of a joint event there and the grand totals outside or the totals outside that's where you can calculate your simple events and your joint events create what we call marginal probabilities or marginal events that you can use to calculate a simple event because if I add this probability of a one and b one and probability of a one and b two will give me the probability of a and we know that the probability of a it's a simple event a one it's a simple event what we also need to realize is the sum of all probabilities should also be equals to zero so if I add everything that is inside the joint probabilities all these four patterns they should give me one or if I add the total whether I add from the roll b one probability of b one plus probability of b two should give me one or probability of a one and probability of a two should give me one as well in a nutshell what I've just said was that probability is the likelihood of something happening the probability should be between zero and one therefore it means the sum of all probabilities as well that is what we just did now should be equals to one and that also states that if it's equals to one if I need to find the probability of a therefore it means to find in the probability of a we'll have the complement which is one minus the probability of b one minus the probability of b just need to write it correctly one minus the probability of b plus the probability of c and that will give me a complement event we also you need to be aware of the following there is what we call an addition rule but normally they will not tell you that in the exam or in your assignment that apply the addition rule you need to just know that this is part of the addition rule so if we need to find the probability of an event a or event b not both of them but either one of them happening that will be given by the probability of a plus the probability of b minus the probability of a and b and that is how you will find the probability of a or b happening but if and only if the the event a and b are mutually exclusive if they are mutually exclusive then the probability of a and b will be equals to zero because that is an impossible event so the probability of finding a or b will be just given by the probability of a plus the probability of b right only only if they are mutually exclusive otherwise this is the general formula that we're going to be using the probability of a or b will be given by the probability of a plus the probability of b minus the probability of a and b we also have conditional probabilities that is the probability of a happening given that b has already happened the probability that event a is happening given that event b has already happened the keyword here is the given part and that is given by the probability of a and b divided by the probability of the given event which is the b so the conditional probability of probability of a given b it's given by the joint probability of a and b divided by the given probability which is the probability of b and vice versa the probability of b given a it's also the same as the probability of a and b divided by the probability of A. And this probability of A and B, it is not the probability of A and the probability of B. This are not the same. This is a joint probability. It's one value. So if they give you the probability of A and they say the probability of A is 0.5, and they give you the probability of B, which is 0.6, and they ask you to find the probability of A and B. Never ever ever say that probability of A and B is that plus that. They are not the same. These two are three different probabilities. Simple probability and joint probability. Always remember that. So if and only if we are given, or not if and only if, if we are given the probability of a conditional probability, let's say we are given the probability of A given B and we are given the probability of B and they ask us to find the probability of a joint event. If they're asking us, find the probability of A and B. If that is the question, but they have given us the conditional probability, then we're going to apply what we call a multiplication rule, sorry, multiplication rule. Which state that? The probability of A and B is equals to the probability of A given B times the probability of B. Now, remember I told you in the beginning the probability of A is given by X divided by N and I said the probability of A and B is given by observation satisfying the joint event divided by N. Only if you are given the conditional probabilities, then you can find the probability of A and B by using the conditional probability times the probability of B. And that is if they haven't given you the conditional probability, therefore we use the probability of A and B as observation satisfying the joint event divided by N. Right? If and only if A and B are independent, if they tell you that they are independent or you can see that they are independent, then the conditional probability of A given B will be the same as the probability of A and the probability of A, conditional probability of B given A will be the same as the probability of B because the given event has no bearing on what happens to the probability that you are looking for. They are independent, so they do not influence one another only and only if the event A and B are independent. Now, you can also use this to prove if event A and B are independent because if I prove that event A and B are independent, therefore it means the conditional probability of A given B should be the same as the probability of a simple event probability of A. If they are not equal, then they are not independent. If and only if event A and B are independent, then the joint probability of A and B is given by the probability of A times the probability of B and that is multiplication rule. Only if and only if the event A and event B are mutually exclusive, then the probability of a joint event A and B for conditional probabilities is given by the probability of A times the probability of B because we know that the conditional probability of A given B is the same as the probability of A and that is that you need to know about the rules. Now, there are additional other things that you need to know when you work with probabilities and one of those things are the counting rules. So, with counting rules, we want to know the number of outcomes possible outcomes that can happen. Number of ways certain things can be done. So, if one of the k different mutually exclusive and collectively exhaustive event can occur on each of the trials, the number of possible outcomes will equals to k to the power of n. Let's say you are running in a race. In a race, there are seven positions that will get a price. What will be the number of ways that you will get a price and that if in a race seven position and only three of you will get a price and that will be seven to the power of three that will give you that.  There are six of them. So, it will be six k. Sorry, k is your outcomes that are one of them should be mutually exclusive and collectively exhaustive. So, there are six signs and the number of trials are the number of times you roll that die and you're doing it three times that will represent your n. So, it will be six to the power of three and that will give you your outcome that is the counting rule. The other rule is a multiplication rule. It states that if there are k one event of the first trial and k two event of the second trial and up to the other k n event on the nth trial, what will be the number of possible outcomes that can happen and that will be given by the multiplication of all the trials. k one times k two times k three times k four up until k n. If I have to go to a park at a restaurant see a movie and there are three parks for restaurants, six movies, how many different ways or different combinations I can do in order to do all this go to a park, eat at a restaurant and see a movie because there are three restaurants for three parks for a student six movies. So, it will be a multiplication of each trial three times four times six that will give me the different ways I can go and visit all three of them or do all three events and that is multiplication the next one is what we call a factorial. The number of ways an item can be arranged or placed in an order and how many number of ways I can take out the book from a library. If there are four books that I need to take out then I can take out four books or three books two books one book at a time that is your n factorial meaning I can take four times three times two times one that is your n factorial on your calculators and factorials at the other side didn't tell you where to find them because you always use them so this is the power you know how to find the power right on your calculator you do have a function some calculator has an n factorial some calculator has an x factorial and that is the function you're going to use on your calculator to calculate n factorial. So, let's say you want to calculate five factorials so that will be five and then you press the n factorial button on your calculator and then just press the equal sign you will see that you will get a hundred and twenty and that's how you will calculate the number of ways you can take out a book from a library so if here I need to place five books on a bookshelf how many number of ways I can do that I can do it four times five times four times three times two times one and that is 120 ways the last one or not the last one the second last one is what we call a permutation and with permutation it also tells me how many number of ways I can arrange certain things or do certain things if order is a priority or there is an order in how I do things or there is a preference in how I do things you have five books and you are going to put them on the three shelves how many ways they can be ordered on the shelf so they also tell me that I need to put them on the three shelves so those are the orders so you will realize that we do have permutation and then we also have a combination so with permutation always remember that there will be order and preference given how many number of ways can you can the books be ordered on the bookshelf so they're giving me an order in terms of how I must place the books so how can I order them on the day so there is an order on this so it means I need to do a permutation on your calculators I've shown you how to do uh factorials but now I want to say you can use permutation on your calculator go and look for npr or npx on your calculators depending so those who are using a case here uh probably it is on you must look for it it is on a multiplication uh function those who are using a sharp calculator it will be on a number six button so how do we calculate npr on your calculator and in terms of in terms of formula you just use your five as your n and your x is your three the smallest values always your x so how do we arrange these books five factorial divide by five minus three factorial which will be 120 divide by two which is six on your calculator you can press five and press the button that relates to the number which will be second function because it's written in orange or shift depending on your calculator shift or second function and then press your npr button and then press three and press equal and and see if you get the same answer you should get 60 if you're not getting 60 let me know and that is permutation with order combination same as permutation without order if you have to arrange five books are going to be selected to read or you are going to or you have five books and are going to select three to read how many different combinations are you going to uh read those books ignoring the order in which they are being selected so now order is not important to use combination also similar the bigger the number is your n so five is your n and x is your three and you just substitute into the formula five factorial three factorial times five minus three factorial and that will give you ten on your calculator also look for ncr usually it's next to your your permutation it's also written as ncr and on your case it will be on a division on a sharp calculator it will be on button number five and that you do the same you say five and you will press second function or nc or shift or shift and then you will press ncr button that corresponds to that then press three and then press equal and you should get the answer of ten if there is no order there is ten ways you can do things and if there is order there are 60 ways you can do the same thing and those are the counting rules and those are the summary for study unit four that you need to know and learn are there any questions before we start answering questions I know that my summary was long but I needed to do that revision so that we are all on the same page when it comes to doing activities Hi Lizzie, so this is the first few sessions but my top calculator is one that you know all the time because it's up in terms of mode I think six to the power of ten for the previous answer so obviously it's quite logical but once we get to more difficult questions I think I'm on the wrong setting okay you still your your cashier is still set on state mode it gives me a lot of options for the stat mode okay so press mode and then press one for c o m p okay and it should have met at the top okay now then that's simple that's fine yeah all right thank you thank you all right so let's if there are no questions or let me check are there any questions on the chat I'm unable to use my microphone okay all right so there are no questions there so let's do exercises exercise one table one below shows fake news media and the type of personality the postings were about which one of the following statement is incorrect so they give us table 1.1 which has social media and traditional media and celebrity politician and superstar and joint event celebrity and social media there are 1800 simple event celebrity they are 2285 now when when you get a contingency table and you get whole numbers like this they know that these are events events whole numbers probabilities there will be decimals decimals or percentages if you get a table and it has decimals and percentages and those decimals are like zero point are between zero and one zero and one or they are between zero percent and hundred percent then you know that these are probabilities if it's whole numbers then you know that they are events so we're going to apply the formulas for the events so now number one it says and we know that the sample space here is 4000 number one says we need to select which one of the statement is incorrect but because we are evaluating all of them it's fine so number one says find the probability of a joint event c and ss so c is this ss is that find the probability of a joint event c and ss remember the joint probability we can find it event satisfying the event the joint probability divided by the sample space is event c and ss can we find the joint event of those two that is your question are we able to find a joint event like for example a joint event celebrity and sm is 1800 is celebrity and superstar can we find joint events no it's an impossible event because they are from the same variable they say category so any person in this instance it cannot be a celebrity and b a superstar they are mutually exclusive so this is correct because they are mutually exclusive what about the probability of c and sm c and sm the event x and the grand total so calculate the probability of c and sm so that will be given by 1800 divide by 4000 and what is that probability it is no point for five okay so therefore that is also correct the probability of c and tm c and tm it's where they both join so that is 485 divide by 4000 which is not point one two not point one two which is correct the probability so yeah we need to find the probability of sm given that c has already happened so we are asked to find the probability of sm given that c has already happened and we know that is given by the probability of a joint event of both of them so i'm going to do both of them s and c sm and c divide by the probability of a given which is c so what is the joint probability of sm and c sm and c is 1800 remember always divide by the grand total which is 4000 divide by the probability of c z is a that is our x so that will be 2285 divide by 4000 and then here we apply mates right which is 1800 divide by 4000 multiply we change the sign we change the division sign to a multiplication and we flip the second fraction what is at the top goes to the bottom what is at the bottom comes to the top and therefore 4000 and 4000 cancels out and we are left with 1800 divide by 2285 and what is the answer 0.787 so 0.79 that is not 0.79 therefore that is not correct we can also do the same with the second one let's do the second one the second one or the last one says we need to find the probability of sm given s s the probability of social media given that the person is a superstar which will be given by the probability of a joint event probability of joint event sm and s s divide by the probability of s s which is the joint event of s s and sm is 515 divide by 4000 divide by the probability of a simple event s s the input event s is 6 6 6 5 divide by 4000 and we can do the same okay so if I flip and change the whole equation 4000 and 4000 will cancel out and I will be left with 515 515 divide by 6 6 6 5 which is equals to so you apply the same method this i did here which is equals to what is 515 divide by 665 not 0.774 so around down to 0.777 which is that and that's how you will evaluate the statement and find the correct answer any questions yes yes I want to ask something yes you can yeah on number one I'm not sure if my question I will put it the way you would understand it I understand that we we made it as mutually exclusive because of um was there another way besides the reasoning by calculation to prove it as this fall under same category that's what I was wondering uh t and sport let's let's get let's get there c and c and s s can they happen at the same time in terms of the information given can a person be a celebrity and be also a superstar no so that will create what type of an event it's an impossible event right yes and an impossible event has a probability of of zero of zero and we know that a a mutually exclusive event is an impossible event right yes and that is why c and s the probability of them happening at the same time will be equals to zero which is mutually exclusive okay now I understand it from the reasoning point of view it's just I was struggling whether if like you get such a question whereby they're in the same category is it possible to make to prove it under calculation or is it by only reasoning even if you want to prove it by a calculation which value are you going to use because you need for example sm and c have a joint event of 1800 where are you going to get the joint event for c and s s yeah that's that's what I was and that's what I was wondering because it was a bit tricky to prove yes I didn't I choose it as zero that it's mutually exclusive but I was wondering if it happens that I get such a question next time yeah so you just need to look at can I find a value that I can use to calculate and if there is no value therefore it means the event cannot happen so the event if the event doesn't happen therefore the probability of you can't even calculate the probability because it's zero the event is zero all right all right some of the questions might look like this where you are given a contingency table but it's not filled in so what we need to do is before we can even answer the questions we need to complete the entire table so let's complete the entire table and then we will look at what the question is asking us so this is students that we surveyed on a question regarding the means of transport to get to school and the following table gives their summary of the results so we have males and females and bar strain and on a car so how many drives oh how many uses a bus 50 50 50 on a on a car 60 100 100 100 and how many are male 95 95 how many they are female 105 105 and what is the total the grand total 200 it's 200 so now we completed the table happy we can now answer the question we're going to evaluate each and every statement right which one of the following statement is incorrect the probability of f and b which is the probability of female and bus so this is the joint event so we need to find observation satisfying divide by n so how many events are female and bus 20 there is 20 of them divided by 200 which is 200 so 20 divided by 200 is not point one it's not point one so that is correct 25 percent of students uses bus to get to school so yeah they want to know what is the probability of a bus and we can multiply the answer by 100 so that we convert the answer to a percentage so that will be observation satisfying divide by grand total how many uses the bus 15 it'll get less of whether they are female or males so they are 50 divide by 200 and we can multiply that by 100 so that we get it to a percentage and what is the answer 25 percent it's a 25 percent 25 percent so therefore that is correct event m and f are mutually exclusive true that is correct because male and female cannot happen at the same time so that is correct the probability of f o o f o o is 0.735 so we know that the probability of f o o can be given by the probability of f remember we have the probability of a or b or we can write it as the probability of a union b which means one and the same thing it's given by the probability of a plus the probability of b minus the probability of a and b now instead of writing a and v's we're going to use the letters that we are given o and f the probability of f plus the probability of o minus the probability of f and o so what is the probability of f it'll get less of what type of mode of transport that will be 105 divide by 200 remember is the simple event which is x divide by n probability of a always going to be x divide by n the probability of a and b will always be x divide by n for joint event on a contingency table so the probability of o 100 will be 100 over 200 minus the joint event f and o how many events satisfies f and o 59 so that will be 59 over 200 so you can say 105 plus 100 minus 59 divide by 200 what do you get 0.73 that will be 0.73 which that means this is correct if a student is randomly selected the probability that a student is a male given that he uses a train so now we need to calculate the probability of a given right i'm gonna move the space here so we need to find the probability of a male given that they are using a train a train is t given that t therefore we need to calculate the probability of male and train divide by the probability of train probability of male and train male and train which is the joint is 24 divide by 200 divide by the probability of train which is 50 divide by 200 so same if we convert this it will be 24 divide by 200 change the sign to a multiplication flip we're gonna get 200 over 50 and 200 and 200 cancels out that gives us one and this gives us one 24 times 50 divide 24 times one is 24 divide by 50 it's 0.48 which means this is incorrect because the answer is 0.48 and that's how you answer the questions so let's look at more questions which one of the following statement is incorrect with regards to experiments using counting rules and assigning probabilities the number of permutations for four items that can be selected from six items is so they already told you they that you need to use permutation so what do we know about permutation the formula is n pr so always remember the bigger number is your n the smaller number is your r four items which is your r six items which is your n so here you will say six p three four why am i using three which is four or you can use n factorial n minus x factorial so depending on whichever one you want to use and the answer is 360 it's 360 so you can use the this one as well six factorial divide by six minus three factorial i need to also give you 360 the number of combinations also here they give you what you need to be using of five items that can be selected from a group of nine so we have five and nine so we need to use n cr which is n factorial divide by x factorial n minus x factorial or you can use it's nine right nine c five on your calculator by pressing nine and then pressing shift or second function and button number whatever the is it the division or is it on number five and then press five and then press equal what is the answer 126 136 126 and in terms of this you will say nine factorial divide by five factorial times nine minus five factorial and it should also give you one 26 and experiment with four steps and five outcomes possible for each step has 625 outcomes and that is a power right so you will use k n remember your outcomes are k k is your outcome n is your first steps so that will be five to the power of four which is 625 which should be equals to you 625 remember your k is your outcome the outcomes of a sample space okay number four in an experiment with 16 likely outcome each experimental outcome has a probability of in an experiment with 16 likely outcome and each experiment will have the probability of so yeah if i'm gonna use an outcome and i'm gonna assume that my outcome my outcome is a and it should be the number satisfying divide by the sample space each experimental outcome so one of them out of 16 what will be my probability not 0.06 what point not six and not not 0.16 a classical method of assigning probabilities is appropriate when all experimental outcomes are equally likely to happen and this is just an explanation of a empirical probability or an empirical event and that is correct because remember if i have a sample space all of them have to have a likelihood of being selected because like if i toss a coin it might land on a head or it might land on a tail unless i'm tossing an unfair coin which maybe the square coin i don't know but if it's a fair coin all events of all outcomes have an equal chance of coming up when they are being when an event is created as well okay so that will be how you answer the questions relating to probabilities so sometimes the question will look like this you won't have a contingency table the probability of a is equals to 0.4 the probability of b complement the probability of b complement is 0.3 and the probability of a and b is 0.2 which one of the following statement is incorrect so this is different to that because this is the probability of a and this is the probability of a and b uh a complement and b complement this is also different so there are two different this is not the opposite of that right a and b is not a complement of a complement b complement they are different so what i will suggest you do before you answer this question you can draw for yourself a contingency table and on that contingency table so they will be four lines i guess you can have a here and a complement here and have b here and have b complement there and there you will have your totals and they will be your total like i said you can take the information given and create your own contingency table so let's do that we know that the probability of a is 0.4 and we know these are probabilities over it so i know that this is equals to one the probability of a it's in the total is 0.4 the probability of b complement is 0.3 which will be here 0.3 the probability of a and b is 0.2 a and b are there so it's here 0.2 complete the entire table i'm gonna give you two minutes to do that i will be back okay are we done completing the table yes yes but give me the numbers so this will be not comma seven not comma seven and here will be not comma six comma six and this will be not comma two not comma two and yeah not comma one not comma one not comma five not comma five because if i add this should give me zero point four if i add this must give me zero comma seven now i've completed the whole table i can just come here and evaluate because i will just refer to all these values that i see here so number one the probability of a complement and b complement is no point one it's not point one so this is the incorrect one because this should be not point one the probability of b and a complement b and a complement and that is correct the probability of b you get less of a's no point seven zero point seven the probability of a given b that we need to that we need to calculate because then we need to find the probability of a and the probability of b divided by the probability of b the probability of a and b is not point two divide by because these are probability already so we don't have to divide by the grand total so the probability of b is not point seven and the answer zero point two eight five that would be the correct one and the probability of a or b complement is given by the probability of a plus the probability of b complement minus the probability of a and b complement so we did calculate the probability of a no we didn't so the probability of a is zero point four the probability of b complement is zero point three and the probability of a and b complement is zero point two and the answer is zero point four plus zero point three is zero point seven minus zero point two is zero point five and that will be correct and that's how you will evaluate the questions you see a contingency table helps with visualizing the probabilities given or events given and then you can answer the questions easier let's use the eight minutes to look at some of the questions so this is one of those questions and they are telling you here in the beginning assume that event a and b are mutually exclusive and if they tell you that automatically some way you should know that if they tell me that then it means the event a and b are equals to zero so they have given it to you and they also give you that the probability of a is zero point three and the probability of b complement is zero point five so now if you have all that information given in the beginning then it complicates most of these things but not you can also do the same draw up a contingency table because now you have all the other information come on so you have a and a complement b and b complement and here we write totals and total and we know that this should be equals to one probability of a is zero point three the probability of a and b they told us a and b which is here they said they are mutually exclusive so that should be zero therefore it means this probability here is zero point three and here it will be zero point seven and what else we are given the probability of b complement which is here which is zero point five then this will be zero point two and this will be zero point five and therefore this will be zero point five easy can you see that easy to complete this and then once you have a contingency table easy to answer any of these questions which one of the following statement is incorrect so we are told that the probability of a complement then b complement is equals to zero is that true it's false that is incorrect the probability of a and b is equals to zero that's what they told us they are mutually exclusive and we can see from here it is correct and here you just calculate the probability of a plus the probability of b you don't need because this is equals to zero so that it will be how you write your answer the probability of a is zero point three the probability of b is zero point five and that should give you zero point eight and that's how you evaluate the next one the probability of b given a you need to know that that is given by the probability of a and b divide by the probability of b and since the probability of a and b is zero that will be equals to zero as well because they are not independent but they are mutually exclusive and you cannot divide the value by zero so you cannot divide zero by any number it will be equals to zero and that will be correct or actually I wrote it all wrong because this should be the probability of a joint event divided by a because it's b given a but it's still the same because it's the joint event of a and b divided by the given so it's still going to remain the same it will be zero because probability of a joint event is zero then the probability of a and b complement a and b complement is zero point three because that is the block and that's how you will answer the questions easy right you just need to go and practice and practice and on that note we have four minutes you can also it looks like it looks like the same question but they are different yeah they're telling us event a and b are independent so now when they are independent what do we know we know that if we if they are independent there are several things that we need to consider the probability of a given b will be equals to the probability of a and the probability of b given a will be the same as the probability of b that is one of the things that you need to consider when you are answering this so similar you can complete the contingency table I think I am drawing it too big let's draw it too small so we know that we have a and a complement and b and b complement we are given the probability of a which is 0 comma three we are given the probability of b complement which is 0 comma five and we know this is one and we know this is 0 comma seven and we know that this is 0 comma five but we don't know anything in between we cannot do anything unless we do the following what else do we know about independent events we know if we have this scenario we can find because they told us that they are independent so we know that if we need to find the probability of a and b we can use either one of them the probability of a times the probability of b remember conditional probability probability of a given b is given by the probability of a and b divided by the probability of b so if I multiply that will be the probability of a given b times the probability of b is equals to the probability of a and b now I know in terms of independent that is the case so therefore it means this probability of a times the probability of b is equals to the probability of a and b and if I know that then I can find the probability of a and b so let's find the probability of a and b we know what the probability of a is is 0.3 we know what the probability of b is is 0.5 so calculate that 0.3 times 0.5 0.15 it's 0.15 right 0.15 so that is the joint probability then 0.15 we can write it there what is the probability of b and a complement and this 0.5 minus 0.15 0.35 0.35 and you can also do the same this will be 0.35 right because it should give me 70 0.70 0.35 plus 0.35 will give me 0.7 so this will be 0.15 0.15 now I have all the information you can come and answer the question it is your homework I have done enough you can also go and look at this you can also answer same questions so yeah you need to evaluate if they are independent you know when they are independent you it's the same if they are independent it means the conditional probabilities of a given b should be equals to a so you can prove that you can prove that that b given c are independent by using by proving that the probability of b given c should be the same as the probability of b or you can do the probability of c given b you just need to do one of them should be the same as the probability of c if it's independent you also need to prove whether a and b are mutually exclusive they told you that that a and b are that so are they equals to zero if they are equals to zero a and b should be equals to zero under the mutually exclusive event impossible event it means a and c the probability of a and c should also be equals to zero can you prove that if you are able to prove it then they will be impossible event event a and b are dependent dependent it means they are not equal so you can use the same the probability of a given b should not be equals to the probability of b or the probability of b given a should not be equals to the probability of b i wrote the first one wrong it should say the probability of a so if you can prove that that is the case then they are dependent a and b are independent you do the same but they need to be equal you you do the same statement but they will have to be equal because if they are independent then then now you need to choose which one is the correct one by choosing any of these symbols that's how complex it is with multiple choice questions you work three times as much as if you were writing a question where you write answers only also this is how they can also ask you questions you don't also not only have to know the calculations you need to know the theory behind every calculation that is why it's very important that you understand the content as well so i've given you lots and lots of work to do if there are things that you are not sure about we can have a discussion on the whatsapp group remember you cannot post the question without showing us how you worked it out here is another example i'm not going to even give you some hint now i've given you enough here is another example where you need to calculate the probability of a and b here is another example you need to complete the table and then answer the question they are asking pay attention to the words given it means you will need to calculate the probability of something given and the given will be that so given will be the neural NP and here they're asking if it's a boy so it's b given p and then you just need to use the formula to calculate remember these are events not probabilities so you will have to divide by the grand total this is another way that they can ask you questions in a statement format if you are stuck let me know how i can unstuck you this is another way i've given you a lot of questions you can see that you can practice and practice if there is things that you don't understand you can ask on the whatsapp group and that concludes today's session our one hour 30 minutes any questions any comments before we close off the session any final word and without any comment or questions or final word then i will just end right here and say thank you for coming through just to recap please remember that you need to understand and learn what the basic probabilities are all about because we're not going to have another session um on basic probabilities until the last session of a week before you go submit your second submission uh so go through the work understand the events understand how to answer them and also remember that you need to also understand the theory behind the basic probabilities not only the formulas and the questions and as you can see that there are so many formulas that we are using and i just summarized all the formulas when i was doing the summary you can also have them somewhere neatly written so that you can always refer to them because you need to know how to calculate the probability of a and b or the joint probabilities or a simple probabilities you need to know how to calculate the probability of a or b which is either and if a and b are mutually exclusive you need to know that the probability will be zero you also need to know how to find the conditional probabilities and in case where you need to use multiplication rule is when you are given the conditional probabilities and you are asked to find the probability of a and b and if even a and b are independent therefore it means the conditional probability of a given b will be the same as the probability of a and you can use that to prove if events are independent or not and that is what we've learned today have a lovely sunday thank you have a great sunday thank you thank you bye


Bella Hadid seems to be one of the most talked about models these days. She's a victorious secret angel, the face of Dior Beauty, and she's constantly appearing in the public eye. She was signed to IMG models in 2014, also winning model of the year back in 2016. And she's only 22. Now, with her father being the iconic real estate developer Mohammed Hadid, and her mother being former supermodel Yolanda Hadid, she definitely had an upper hand in her rise to the spotlight. Seeing as her family also appeared on the show The Real Housewives of Beverly Hills. We all know that Bella is a gorgeous girl with a rise in career as a supermodel, but in this video we're going to be focusing just on her beauty transformation over the years. Although Bella strongly denies rumors of getting any work done, I'm not so sure. Let me know after this video what you guys think. What's going on guys and girls, my name's Kara and this series we've titled Before and After. We're going to talk about Bella's style, look, and any rumored cosmetic procedures. I've done other celebrity transformation videos on the lights of Ariana Grande and reality star Farah Abraham, and we'll have links to those at the end of this video. If there's anyone else who wants me to document, let me know in the comments down below. Now let's get into this video. Like I said, Bella has denied the fact that she's gotten any cosmetic work done. Even her mom Yolanda stepped in earlier this year addressing the rumors.  This triggered some commenters to inquire about her daughters and why they got work or fillers done, namely Bella. Yolanda's response, none of my children have ever done fillers or botox or put anything foreign in their bodies. They know better after seeing what I went through. Bella also told InStyle last May, people think I got all this surgery or did this or that. And you know what? We can do a scan of my face, darling. I'm scared of putting fillers into my lips. I wouldn't want to mess up my face. All of this may very well be true and it's no doubt that Hadid girls were always beautiful. But I think photos tell a slightly different story. Looking at Bella and her teen years around 2012-2013, her face looks a lot different. Firstly, Bella's nose. This is the most noticeable change in my opinion. Before Bella's nose was a bit larger and not as sleek. These days her nose is smaller and much more refined. This could have been from a nose job or even fillers such as a nose tip refining procedure where fillers can be used to lessen any bumps and make the tip of the nose more raised. Considering how different Bella's nose looks, my guess would be a surgical nose job. But who knows, right? It also looks like Bella's eyebrows became raised and more arched over the years. This could very well have been the work of fillers, which would have helped to create the wide-eyed look she has today. Bella's cheekbones are much higher up on her face as well. Bella has lost weight over the years, so this could have caused natural changes in the shape of her face. However, fillers being used here is another possibility because her cheekbones are much more prominent than before. Finally, Bella's lips are definitely fuller. These days lip fillers are so common one probably wouldn't even think about it. When you compare older pics of Bella with how she looks now, her lips do look bigger. Remember, the star deny is getting any work done, so these are all just ideas. Moving on to Bella's body, it doesn't really look to me like Bella has gone any work done here. Standing at around five foot seven, Bella has a typical supermodel figure with an exceptionally small waist. She's certainly lost weight over the years and even admits to her weight changes. In an interview with People magazine back in 2016, Bella said, My weight fluctuates so much. I really didn't mean to lose weight.  My weight fluctuates and so does everybody's and I think that if people are gonna judge, that's the worst you can possibly do because everybody is different. At this time, she was prepping for her debut on the Victoria's Secret runway, which obviously one has to look their best for. Bella claims that she was following a strict diet and exercise plan to keep her body extra fit, which included eating lots of protein and working out for three hours daily. One hour's enough for me, but if I was gonna be a Victoria's Secret angel, I would definitely be doing the same. In her daily life, Bella says she drinks a lot of water and tries to eat as much healthy food as possible. She does admit to cheat days, saying she'll still enjoy carbs like pasta, pizza and bread. Nutritionist Charles Passler worked with Bella in terms of keeping her diet healthy and optimized, and he claims that her plan includes a good amount of proteins, healthy fats, and lots of non-starchy vegetables. It seems that balance in eating healthy is the key to Bella's slim figure. She probably eats healthier than a lot of us, but still likes to indulge when she can. And plus, she has the motivation of being a Victoria's Secret angel to keep her in check. Looking at Bella's hair and makeup style, she seems to be content being a brunette. I think it's her signature look, seeing as her sister, Gigi, is known for being a blonde. This definitely makes them each stand out in their own way. They say what they're gonna say, have a drink, drink, found the bad bite, life is like me, the heart to come by. I think Bella looks great as a brunette, and it really brings out her light eyes and complements her skin tone. Bella's makeup look is pretty natural and minimal most of the time, and I think it's what looks best on her. She doesn't need much or really anything to make her look good. Not to mention, her skin always looks clear and flawless. When the star does wear more makeup, she seems to go for black winged eyeliner and a nice lipstick. Being a supermodel, it's pretty much a given that Bella has good fashion sense. Whether it's on the runway or her everyday wear, Bella is a trendsetter. She's evolved since the early days though, and when she first broke onto the modeling seen as a teen, she had a more casual LA girl style. Over time, Bella has become a lot more daring with her fashion choices. I feel like she was one of the first responsible for bringing back those tiny sunglasses from the 90s too. There are way too many Bella looks that I like, but some of my faves over time include the red gown she rocked at the Cannes Film Festival a few years back, as well as a couple looks she's worn on the Victoria's Secret red carpet. A lot of her outfits are pretty sheer and revealing, so you certainly have to be brave to wear them. Looking at Bella's street style, she always looks great. She seems to like mixing high fashion with trendy and casual pieces that still show her youthfulness. Not to mention, Bella always looks great in a bikini. Bella's still young, so I'm sure she'll evolve even more over time, but she's already come a long way. She's gone from having a style much like your average teen girl to a high fashion supermodel within only a few years. Although Bella claims she's never gotten any work done, it's a bit suspicious. Her face has changed a lot in a short amount of time, but either way, I think she looks gorgeous. All right guys and girls, that wraps up this video here on Before and After. If you wanna talk more about Bella's transformation, you can hit me up on Instagram. Let me know who else you'd like me to make a Before and After video on, and I'll see you guys in the next video. Bye.


at our Tezal course in Barcelona we endeavor do everything possible to ensure that you will have an enjoyable and rewarding time in the wonderful city of Barcelona. Our Tezal course center here is home to a small and friendly team of staff which offers all course students a warm welcome and a wide variety of social and leisure activities. Our staff can arrange evening excursions to the top as bars and cafes, sailing trips and other watersports activities, skiing, in winter, wine tasting and trips to many beautiful beaches. We arrange a welcome dinner at the start of the Tezal course in Barcelona to help you get to know the other students on your course at an early stage. The Tezal course is intensive but we do everything possible to make sure that you ample free time to make the most of your time in this lovely part of the world. Barcelona is perfectly located between the warm waters of the Mediterranean Sea and the Tibor d'Abo Hills. The idyllic Balyric Islands are to the east of the city, the famous resorts of the Costa Brava, Wild Coast and the towering mountains of the Pyrenees, are to the north, the wine growing region lies to the west, and the lovely Roman city of Tarragona and the beach town of Sitges are to the south. There is something to door somewhere to explore whichever way you turn in this charming city. Barcelona is home to the works of many of the most famous artists and architects of recent times, including the modernist spires of Gaudí's Cathedral, Picasso's Cubist art and Dali surrealism. The distinctive Catalan perspective has had an impact that has felt way beyond the region. You stroll down the most famous street in Barcelona, the Ramblas, which leads right down to the sea, you will witness grand opera houses, beautiful churches, street artists and mimes, cafes and bars galore, and of course great shopping at and around the market. The centerpiece of the Gothic Quarter is its fine cathedral. The Picasso Museum, which houses the best collection of Picasso's work in Spain is nearby. It says justifiably also famous for its nightlife. It sometimes appears that every other building is a club, cafes or bar. Nightlife is Spain and in Barcelona in particular tends to start and finish late and dinner is usually served after 9pm. After dinner you can pick from a huge range of bars or cafes to continue your evenings, entertainment and those with plenty of stamina can then move on to the open air nightclubs in the Tibor d'Abo Hills amongst other options. With all this to offer, Barcelona is a great place to take your tesal course as well as having a great time outside of course hours. Whether you like beaches or mountains, art or history, cafes or nightclubs, Barcelona really does offer something for everyone.


The top 5 places to teach English in Thailand. It is no secret that Thailand is one of the most popular Tefl destinations in Asia, and indeed the world. Foreign teachers have been flocking to the country for many years to enjoy the fabulous mix of tropical weather, fascinating culture, friendly local people, and a low cost of living. Most importantly, there are also plenty of jobs available in many diverse locations across the country. For an insight into the most popular options, take a look at this list to see which might suit your lifestyle the best. 1. Bangkok. Thailand's capital city is unsurprisingly home to the largest market for foreign teachers in the country. Across this vast city, you will find countless job opportunities in government schools, private language schools, international schools, colleges, and universities. As a major world city, the cost of living in Bangkok is higher than you will find elsewhere in the country, but you can also expect to earn a higher salary to compensate for this. Although it can certainly be a hectic place, especially during rush hour, the city is relatively easy to navigate due to the modern BTS sky train and the underground MRT systems which are clearly signed in English. The main attraction for many teachers is the city's mix of local culture and modern international living. Depending on your mood and budget, you can choose to dine out on incredible street food for just a dollar or two, or you can splash out on a swanky meal and a night out at one of the city's many trendy clubs, bars or music venues. You will also find plenty of great shopping options, from huge malls to traditional markets. With its wide range of modern amenities, Bangkok might be the right choice for you if you are new to international travel, used to big city living or simply fond of your creature comforts. 2. Phuket. This Thailand's largest island, Phuket has rapidly grown from a few tiny fishing villages into the country's premier resort destination in just a few years. It is now home to a large expat community from all over the world, as well as an ever-growing local population that is drawn by the largely tourism-based job market. Across the island, there are many job opportunities on offer for foreign teachers, although you can expect a relatively high level of competition for the best positions. For those who do find a suitable position, the reward is to live on a tropical island that manages to balance city life with a laid-back beach vibe. Among its many charms, Phuket offers plenty of traditional temples, markets, and colorful festivals, as well as modern shopping malls, multi-screen cinemas, and water parks. Of course, the island's many fantastic beaches are also a huge attraction. Around the coast, you will find beaches to suit old tastes, from peaceful palm-lined sands to lively resorts complete with water sports and a vibrant nightlife. While some parts of Phuket have become a little overdeveloped for some tastes, you are only a short boat right away from several other stunning islands where you can get a feel for the region's outstanding natural beauty. If you are looking for a location that mixes modern conveniences with a relaxed lifestyle, Phuket could be your dream destination. 3. Chiang Mai. Thailand's revered northern capital offers a vastly different option from the hectic pace of Bangkok and the tourist magnet of Phuket. Due to its relatively high elevation and location in the north of the country, Chiang Mai enjoys a considerably cooler climate than many other places in the region. There are also plenty of public and private schools to be found, although salaries are generally lower than some other areas. However, this hasn't stopped its popularity among foreign teachers, as many are prepared to earn a little less in return for all the benefits of living in Thailand's cultural capital. Although Chiang Mai is many miles from the nearest beach, it is surrounded by a backdrop of stunning proportions. In all directions, there is natural beauty to be enjoyed, including jungle-covered mountains and spectacular river valleys. The area around the city provides the perfect playground for enjoying a wide range of adventure sports, as well as some of the best trekking to be found in Southeast Asia. Within the city, you will also find many cultural highlights, such as ancient temples and handicraft markets, as well as cool coffee shops, bars and cafes. If a mix of nature, adventure, culture, and nightlife is what you are looking for, look no further than Chiang Mai. 4. Koh Samui. The beautiful island of Koh Samui is second only to Phuket in terms of size, but it has yet to be developed to the same degree. The island has plenty of government schools who employ foreign teachers, as well as a few international schools aimed at both the thriving expat community and wealthy ties. If you can secure a job at an international school you should earn enough to live very comfortably. However, as an island, the cost of some basic necessities is higher than on the mainland so those with lower paid jobs might need to be a bit more frugal. Koh Samui offers a real mix of locations, from quiet, rural districts to built-up areas with a busy nightlife. Wherever you choose to live you will never be far away from a beautiful sandy beach where you can relax during your time away from the classroom. Whether you prefer to swing in a hammock under the palms or tackle something more energetic, such as surfing, diving, or snorkeling, there is the perfect beach for you. If you would like to experience the peace and tranquility of a tropical island, that also has the option of a big night out, Koh Samui is certainly worth considering. 5. Isan. If you are looking for a more authentic Thai experience in an area that has not been developed simply to please the tourist market, then the northeastern region of Isan might be perfect for you. This huge area is home to several medium-sized cities that offer plenty of opportunities to teach in government schools. While you should not expect to get rich working in this environment, the local cost of living is the lowest in the country so you should still be able to live well on your salary. The majority of teaching jobs are found in the four main cities of Korat, Khon Kain, Uban Ratchatani, and Udon Thani, all of which provide a great base for exploring the region further. The obvious attraction of the Isan region is to experience Thailand in a way that most tourists, and indeed teachers, never do. The area is home to a third of the country's population, yet very few tourists head here. The rural parts of the region enjoy a slow pace of life that seems a million miles from the hustle and bustle of Bangkok, yet it is only a one-hour flight away. There are several unique national parks to explore, as well as the mighty Mekong River which forms a natural border with neighboring Laos. Isan is also famous for its local cuisine which includes two of the most popular Thai dishes, lob, and som tam. If the authentic side of Thai life appeals to you, why not take a look at the Isan region? Are you ready to teach English in Thailand? If you want the best possible start to your teaching career, why not complete your TEFL certification course at one of our training centers in Thailand? We have popular centers in, Phuket, Bangkok, Chiang Mai. You can also find out more about living and working in the country via our Thailand Country Guide.