And it's Idil Kabeer today, and we have a number of stories across Nigeria. President Mohammad Abu Hari wants citizens to be security conscious, even as Muslims celebrate Idil Kabeer. He spoke to journalists who met him in his hometown in Daorah of Katsina State, where he had gone to pray and observe the Eid. I think the Emirates said it all, and those who address the people, they should be security conscious, those that are distracting the attention of governments at all levels, local, state and other, they live among them, they know them, let them expose them so that we can have peace in the country.


Hello and welcome to First-Rate Tutors Knowledge Hub, your channel for everything you need to learn, whether you're a student in school or university, or if you're simply someone who's just curious and you want to be better informed. Now, there have been many people who have said the coronavirus pandemic is a modern great plague. Many have drawn parallels between COVID-19 in 2020 and the great plague which occurred in 1665 to 1666. Is this really the case? Watch the video and discover more. Now, Samuel Peeps in his famous diary entries stated, the plague growing very raging and my apprehension of it great in July 1665. And this has some haunting similarities with COVID-19. Indeed, the spread of COVID-19, also known as coronavirus, has led many to draw similarities between this pandemic and the great plague. Now, before we go into this, it's really important to distinguish what the great plague was versus coronavirus. Now, the great plague which broke up between 1665 and 1666 was a bacterial infection which was found in rats. It transferred from rat to human because of fleas which bit into infected rats, then bit humans and passed on the illness. However, COVID-19, also known as coronavirus, is a disease caused by coronavirus, which is a virus that affects our respiratory system. It's likely passed on from animals to human and it's alleged that the virus emerged at the Wuhan live animal market somewhere where someone ate an infected creature allegedly and this of course was in China. Now, it will be really interesting to first see how the COVID-19 pandemic really spread. So, as I've mentioned before, it started in China in mid to late 2019 and it merged from Wuhan's wet market allegedly. Then between January and March 2020, COVID-19 spread across Europe with significant fatalities in places like UK, France, Italy and Spain. However, USA was not spared. Indeed, on January the 21st 2020, USA registered its first confirmed case and COVID-19 subsequently spread. Globally, developing countries in places like Africa, Latin America and Asia have also been affected, some severely like India, whilst other places like Kenya have registered COVID-19 deaths but seem to be containing it relatively successfully. Now, let's think about the Great Plague. So, the Great Plague of 1665 arrived in London on Dutch trading ships from Amsterdam that were carrying bells of cotton. Black rats were on board and some were infected with a bacterial infection. The plague was spread by fleas which fed on their blood and once they bit an infected rat, the infected split spread the disease to humans by then biting human beings. Now this Great Plague began in London and the poor overcrowded parish of St Giles in the field. It started slowly at first but by May 1665 43 people had died. In June 6,137 people died and in July 17,036 people and at its peak in August just a few months later 31,159 people had died. The plague also spread to many parts of England and York was one city badly affected. Incubation took four to six days and when the plague appeared in a household the house was sealed condemning the whole family to death. There was even a watcher to ensure that nobody left for 40 days. These houses had a painted red cross on their door and the words Lord have mercy on us. Also, there was no requirement to report a death to anyone so, searches of the dead were chosen by parishes to go and look for corpses and work out what they died of. Also, plague doctors were employed to treat people who had the plague. By 1666, 68,596 plague deaths were recorded in London meaning between 30 to 50% of London's population died of the plague. Now are there outbreaks similar? So when you think about COVID-19 versus a Great Plague, there are some similarities. Firstly, of course, both are transmitted by animals. Also, both of them, if you think about the areas that they really vastly affected actually had been imported. In other words, if you think about COVID, the areas in the regions which have been worse affected than China such as Europe, USA and so on had this of course imported from China by travellers and of course London and the UK which was heavily affected during the Great Plague had imported this from of course the Netherlands with the ships. Also for both diseases, incubation usually is five days. Of course, we know that it can be give or take five days but actually for both of them very commonly the incubation period lasts for five days. Also, of course, both diseases thrive in cities especially highly populated cities that then can infect and go between lots of people very easily. Also, both share some similar symptoms such as fever and coughing. The bodies of the deceased, in other words the bodies of those who've sadly passed away for both COVID-19 and the Great Plague are highly infectious. These bodies are really, really infectious. And also, of course, an effective way to tackle these diseases is through social isolation. So people being isolated of course in the case of COVID-19 were seeing lots of lockdowns in places like the UK, Spain and Italy and of course in the Great Plague, as I've mentioned before, once a household was infected it had to be locked for 40 days. However, there are some significant differences between the two. Do bear in mind of course that COVID-19 or coronavirus is a viral infection whilst of course the Great Plague is a bacterial infection. A virus and a bacteria are significantly different. Also, there are other symptoms which are highly different in the two illnesses. In COVID-19 the symptoms include fever, a severe cough, fatigue, loss of appetite, loss of smell, body aches. Whilst symptoms in the Great Plague involve having large blisters, buboes, chills, muscle cramps, seizures and intense pain. Also another difference to bear in mind with COVID-19 is fortunately a lot of our countries have better public health systems whilst of course during the Great Plague London and other key cities across England were really unsanitary, overcrowded, lots of open sewers. Also of course COVID-19 is being tackled thankfully by qualified doctors and nurses whilst in the case of the Great Plague it was plague doctors who looked after that and whilst when it comes to coronavirus, public health officials and morgues are dealing with those who've passed away. When it came to the Great Plague, public health officials and morgues didn't necessarily deal with them, it was essentially body watches. So that's all, if you found this video useful do consider giving the video a thumbs up and also consider subscribing to our channel. But also make sure you visit our website which is www.firstreachutors.com. As we've mentioned we are a knowledge hub and we specialise in empowering students when it comes to any academic discipline so make sure you check out our channel and also check out our website to find out more information to download lots of useful study materials. Thank you so much for listening.


Hey Psych2Goers, welcome back to our channel. Have you ever heard of a dark empath? I'm sure you've probably heard of an empath, someone who ranks high in effective and cognitive empathy. They have an acute sense of what others are feeling and thinking, but dark empaths don't necessarily rank high in effective empathy, but cognitive empathy, while also displaying dark triad traits. The dark triad is a term used to group three malevolent personality traits in psychology. The dark triad comprises of the traits of Machiavellianism, Narcissism, and Psychopathy. Those who are part of the dark triad personality type rank low in empathy and high in the three dark traits. An average score may mean you rank low in the dark traits and average in empathy, while empaths are ranked high in empathy and low in the dark traits. But a dark empath ranks high in both dark triad traits and empathy, but just because you display empathy doesn't mean you often share another's feelings emotionally. That's effective empathy. Dark empaths can often use cognitive empathy to their advantage due to their dark traits, meaning they know what it is you want and feel, but dark empaths may use it against you. There are plenty of reasons being a dark empath may have its advantages. Possessing a great deal of empathy and ranking high in dark triad traits, such as psychopathy, could be very useful in certain occupations. Surgeons, soldiers, first responders, and others in intense and high demanding jobs may find it useful to have a bit of cognitive empathy, understanding others, but being able to shut off absorbing others' emotions and relating so much that you often need to recharge. Seeing other people's motives and desires helps them make important decisions, and although they understand others' feelings, they have the ability to choose not to be empathetic if needed. They may even have some secrets up their sleeve based on the unique combo of empathetic and dark triad traits. So what are some of those secrets? Here are five secrets dark empaths don't want you to know. Number one, they know how to get the right reaction from you. Because empaths are so in tune with others' emotions, they can learn very quickly what strongly touches their emotions. Maybe a certain subject subtly gets on your nerves, or perhaps a certain someone or topic ever so slightly brightens your mood. Many dark empaths can easily pick up on these emotions and in turn know exactly what to bring up, what to say, or what to do to get this reaction out of you. They've simply observed how you react and know exactly what to say. This isn't to say dark empaths aren't genuine in what they say or do. Their sensitivity to your desires and feelings just gives them a boost in predicting how it is you'll react. Number two, they often know how to act by watching films and television. It's being noted by psychologists that some dark personality types obsessively watch TV and films, observing normal human responses for emotional mimicry. Basically, they want to further develop cognitive empathy. Since they rank high in cognitive empathy, they may know just how to act in the right situations, but sometimes it may feel forced to others. This is because their effect of empathy ranks low. A study by day city and associates found when subjects high in the dark triad trait psychopathy imagined pain to themselves, brain regions including the anterior insula, anterior mid-singulate cortex, right amygdala, and somatosensory cortex, displayed average response to pain, implying they are indeed sensitive to the thought of pain. But when they imagined pain of others, these brain regions did not become active. Number three, they can often tell when you lie. Since dark empaths and empaths alike can notice the subtle reactions and changes in moods and others. Some can often tell when your emotions don't line up with what you're saying. Nobody knows what someone is exactly thinking. No one's a mind reader, but empaths are often more sensitive to mood changes and the little subtleties in others' expressions. Knowing when someone is lying means they could or could not use it in a negative way. Depends on how high they rank on dark triad traits. Number four, they know how to get what they want. Because dark empaths can read others easily, they know what to say to get what they want. For example, on a date with someone, a dark empath may pick up on their date's subtle desires, pay attention to their reactions, see what they respond best to, and alter their words and actions to fit exactly what they want. Even if this doesn't fit who they really are, they simply know how to get what they want. And number five, they can manipulate you. Dark empaths can often use their high level of cognitive empathy as a means of manipulating others. Often noticing your subtle reactions and feelings, they may just know exactly how to use your emotions against you. Some dark empaths have a hard time with boundaries and often focus on fulfilling only their needs. If you notice the end of every argument or discussion, their needs are only met. They may be manipulating you. This isn't to say every dark empath wants to manipulate others, but if they needed to, they wouldn't have a hard time. So have you ever heard of a dark empath? Let us know in the comments down below. If you wanna hear more of the dark empaths, check out our other videos on the topic. If you found this video insightful, don't forget to click the like button and share this video with a friend. Subscribe to Psych2Go and hit the notification bell icon for more content like this. And as always, thanks for watching.


Have you ever gone to an event or a party where you meet somebody and they just walk in and you can tell that person is a quote winner. They have a big smile on their face, they're super personable and when you think about them you're probably thinking that they're someone who did not have a very hard life. Like they just give off this feeling that life must have been always easy for them. In this video I want to share a very very clear and interesting kind of mindset that I've noticed having coached hundreds of people. The one fundamental mindset between high performance and a life that continually stays the same. What's up guys? Alex Hine, author of the book Milk the Pigeon, a field guide for anyone lost in their 20s which you can get on Amazon or Audible right now. Now I've included a free link below this video which is for a free worksheet to plan out how to have the best year of your life. So if you're learning how to set effective goals check out the free link right below this video. So the first belief, the fundamental belief that keeps your life guaranteed the same. The belief that allows you to still continually disappoint yourself is this idea that things are just going to change and nothing that I do will affect that. It's the belief in love that you know there are no good girls out there all the great ones are taken everyone else is interested in me for the wrong reasons or they don't like the real me. It's the belief that in work it doesn't matter how hard I work at my job I'm never going to get a promotion everything's rigged and everything's unfair it's just the economy it's just the actual business I'm in the field I'm in things are never going to change or it may be that there's something you believe about yourself to be fundamentally flawed or some trait that you believe is genetic whether it's your weight whether it's being a little bit depressed or a little bit anxious some thing that you have that you believe about yourself that is fixed and will never ever change now what I can guarantee having coached people and having worked on this myself is that if there is any belief that predisposes you to having the same life year after year it's the belief that nothing that I do will result in any change in my life because those things are out of my control so now on the other side what's the belief that leads to your life getting better right you're probably watching this because you want something to change or something to get better the fundamental belief that whether or not the person is conscious of or not is that the things I do that I will use to figure out to get a better life I can figure it out I will actually be able to figure out how to be financially successful how to get the person I love how to be in a great relationship how to get fit how to have peace of mind or how to deal with my anxiety or depression the fundamental rule it's a belief whether or not you're conscious of it or not whether or not that quote winner that you've observed somewhere is conscious of it or not is they have a belief in self-efficacy that the things I do I know I can figure it out I know there's a way I know that whether it's by googling it or consulting the top experts there's a way I can figure out this problem what they call now a growth mindset presents itself in love in the sense that if I get rejected by five girls in a row I'm not blaming it on women I'm not hating women I'm not hating the state of dating I'm not hating texting it's me right the common denominator in this failure is me so what can I do about myself to improve myself to improve the way I approach this problem to actually get a better result next time and if it is my career like look bureaucracy all this bullshit and work is a real thing so I can sit here and complain about why I'm getting passed up for jobs or why I work harder but because it's a corporate environment even though I deliver 10x I'm not getting paid 10x I can have the fixed mindset and complain that no matter what I do it's not going to make any difference because I work at this life insurance company or I can figure it out I can take the initiative to try to build a new separate part in that business and talk to my manager or talk to my boss I can do what I did which is I said screw corporate I'm going to become an entrepreneur so it's this just a fundamental philosophy that I'm going to figure it out rather than resigning oneself to just mediocrity in the sense that believing we can't control anything and whether it is something you perceive to be genetic or not maybe it is your weight is one way you're too thin and you want to be a big guy or you're too big and you want to be a thin girl whatever it is if I say this is just genetic this is just how my parents were and every time I try I always fail so why bother trying well that's guaranteed not going to get you anything that you want in your life so I can either resign myself to mediocrity and staying the same or I can try to have this growth mindset and believe that I'll figure it out no matter what happens but the tricky thing the plot twist is that it just begins with that belief and you don't even have to necessarily believe that belief to begin if you just try it for a hundred days try what happens when you think I can probably figure this out and just watch what happens in the action you take all right guys so before you go there's a free goal-setting worksheet down below this video it'll help you figure out how to plan out and have the best year of your life this coming year in 2021 and then before you go I have two related videos right here


i'm nervous i'm so nervous for you i'm for sure i'm for sure just for two years yeah two years i know for a good moment i'm not lying i think i'm checking i will all right i'll call you in a bit all right it's checking time so i'm currently in the middle of studying for my final oski exams in medical and i haven't been able to sleep at all last night for the first time in like forever because today is the day where we get placed in our hospitals for the next two years of our life as a junior doctor so i woke up to an email which said foundation program allocation i'm gonna check it now i'm very nervous i so much rides on this on this result it determines so much of my life over the next two years and so i just wanted to go well uh i hate i hate clicking result things in emails man the person who it's most important to know right after me alexia you have been elected to the following north london oh my god north london north london so such a huge relief such a huge relief oh we studied hard we put in a lot of work and we definitely had some luck as well a lot of things went in our favor clearly for this to go well um i'm just feeling very thankful very very thankful very lucky a lot of gratitude right now


What's up everybody? Welcome to SPM Bars. My good name is Miss Kethinge. I'm hanging out with the staff for today at the Chipukizi live recording. It's the first one and we can't wait to see what he has in store for us. But he is here with me. Nataka puskiakama kua na butterflies in the Stomaq Chipukizi. Mimimimi. How are you? I'm very good. Go to see. Go to see you too. Noski awa. How do you feel? Nasi kiawe? Bana nasi kiapoa. It's a beautiful day. God has been good to us. Nana fri that tu me manaji kufika kunaide. Tu nasa machi pukizi, she was live recording again. Again. Amazing. Let me ask you because it's been a long time chipu kizi before you know you probably went on stage and did something of your own. How do you feel? Fun enough. I always feel butterflies. Even when I'm going for meetings. So of course it's a mixed emotions. I've never been so happy and scared at the same time. But I think it's not been easy to get here. I've had, let me tell you, for the last 10 years I've been in the industry. I've gone through it. And one of the things that has happened is that I've been able to learn and study. And so I know the dynamics. Like for example this is the third time that I'm producing a TV show. So the mixed feelings is from experience. The mistakes I made from my previous shows. How am I going to improve? How is it going to come out? But I know at the end of the night we're going to have a great show. For sure. We had a very big show planned a few, like a month ago, two months ago and then you fell ill. You had to reschedule, postpone it. How was that moment for you? Because it was big.  How was that moment for you? I think it was a very interesting moment. But it was one of those moments that I got to appreciate life. Because when I got healed there was nothing else I could do. And yet I had so much expectations myself. But then I realized that we are not in control. But we tend to chase the day to day life. We forget about just taking care of ourselves. That being healthy and being alive matters more than just being famous and planning to do shows. Making ten million is good but also living 100 years is more better. So I learned to prioritize. It was a learning lesson really but I didn't take it with bile. I didn't want to... God has his own reasons. And for sure elek chunataka kuambia at lazima to prioritize on ourselves. We are the price. Your health is your wealth. As long as you're not don't overwhelm yourself trying to chase success. Kita ingiana tu. How are you doing now inu health wise? I'm sure you did not really tell us what was the problem. I'm not sure if you really want to share with us what had happened but how are you doing right now? Right now because I don't remember what happened to me. I feel so healthy. I have had stormak issues for the last two years and I did not take them seriously. I did not stop and go to hospital to get checked. I was just trying to self-medicate myself. By the time I realized it was when it was critical when I had that moment is when my health ignore. So now I'm good, I'm better. I wake up for the first time na amukana siyaga mamutubana. For a very long time and me I thought it was a normal thing that I thought it's Alsace. Marafikizangu ingina wali kuwa na sema ni Alsace but it was not. It was other issues but I'm happy now that Kosao. I mean we can see Kosao, Jutumonu Merudikazi and even now you're doing this live recording that means you're doing well. Now this comedy show is going on KBC. Why KBC? That's home. We usually say it's Kenyan home in Binadamo. So this is a home of talents. All the great talents Leonard Mambo Botella Mehid B. Jeff Koinange Mehid B. Churchill Mehid B. Mama Kayai All the great artists started there but also I'm looking for a stable platform. You know this thing of always listen, we keep on saying that it's okay social media is doing well. As Piamba zina uwa mbaka na yona state house it's okay Kenya wakenya niwale wale wakenya wakukwa TV wakenya badawa na watch, wanna consume mainstream news. So then now that KBC and Kenyan government has a gender to push talents I feel like it's easy for my talent to be amplified on KBC because then they want to help talent monetize their talents. Most of these other stations so I feel like for now and in the next 10 to 20 years KBC is going to stabilize again and I want to be a part of the success story when KBC gets back into his old glory. And then also the people who watch us I didn't go and be here my dear wale watu mbawa na to support the majority of these Kenyans niwatu niwase wako home machinani niwase wako ghetto free to hair like this like that kuna wase kwa fod kulipia easy TV Zenuza kizungu daily lakinnik KBC wuku amtani daily so see it tu naka kupiawa kwa kwa kwa kwa amtani na wala kushindo haku access entertainment tu is not easy to watch these other stations kusaba bawa ezi li piye ki tu ya ezi ma dikoda weekly o monthly or whatever kamu o kuna report umtani na waka ABC kwa kwa kwa kwa kwa sumon Afingirsty kwa kwa kwa kwa kwa bumbe wanshy apo wana kama kwa kwa pataswatapu na epic wako wambia Uko inja nikubaya. Washi apu upate iyo. Kozinao, sisi wase wa shagz. Liyanda usyago jyuzin kona wa sayona tumiyamuli kamuizi. Niku apu usyago una patana struggle. Niku kwa shushu yungu. Nuna shushu yungu ya na watch 2KBC. Nuna. So, nikariyalize, tunaza kuja wuku tujifanya tuko biya sharani. Lakin ni, the real audience, the real fans, our people, wale wako holokalau, bungoma, kakuma, lamu, all these other spaces, they can easily watch us there. But that I mean the people who, you see, there is the industry, there is the people, there is the industry people, and then there is the people who support us. Those are our target people. Natu na ju awana rizona tvisurina KBC. So that's why. Okay, so it's going to be going live, it's going to be airing on Mondays, what time? It's to 9pm, and I think, but we will make the announcement. It's also going to Y254 TV. We will find a day where, you know, once we air it on KBC in Andakwa, Y254, yeah. The other day you posted cha-chil and you said the best thing you can do to someone is give them a chance.  Have you given someone a chance and who are those people that you'd like to say, I've given them a chance and I hope they can make a name for themselves. I don't even have to mention their names, but even off camera, we have given, that's what I do. I give people chances. You know, sometimes even giving someone a chance is not giving them a platform. It's listening to them. When somebody stops you and say, ni aji bro, ko ajin, ko fiti, manze mi na nafkiri naia idea, that's all it matters. There are people out here who are going crazy because they cannot find anybody to even listen to them. So the best thing you can do to anyone is give them a chance, you know. Chibukizishio itzel. The reason why I'm doing it, let me tell you. Maybe because of how the beautiful things that come with success, maybe somebody will say, and people have told me that, maybe I don't need to be back on TV. Maybe I don't need to be doing these things. Why am I backing to do these things? You know, you could do these other things, but because I know I might be that one interview away from that one person who is in the village who wants to make it. And that's what one person is what I'm looking for. That's what inspires me, that if tomorrow when we wake up, that yes, leo tutalala master, lakin itu kiamkakeshubana, labda tunapapasia, umtumoodya tuha jekwetu, kisha mambuyaki iwemzuri. That's why we exist and that's why we have Chibukizishio. And that's why I maintain my name. I'm always upcoming because I know there are so many other people like me. So I'm always looking for people who want to make it. So I'm always looking for people who want to make it. So yeah. Hey, as I finish up because I need to let you go to Chibukizi, you have been in so many events interacted with so many people the past two weeks. I just want to send this idea here. What has been your biggest highlight? For the last one week, wow, highlight. For two weeks, leo tutalala. What has been the best highlight? Selasia. Selasia, you know what? Selasia, weki. Yeah, I know that. What has been my best highlight? What has been my best highlight? Wow, that's a very... I don't look at them as highlights. Okay. I think the whole week has been a highlight. Not everybody I've met this week may it be Selasia if it's the week I was born. I got to be in the same room with four first ladies of different countries, of different African countries. And when I was in that room, when they were going around... So yeah, we were at this tree of hope, you know, part of the... And so now the first lady walks in with four other first ladies. Yes. So nika jipata ni koapo na it was quite juicerson liya za kufikria manze kwa village manze. Nikanda kwa mbia makuzozangu manze that ni mewit na first lady wa a certain African country. Inasmuch as it's neither here nor there. But for me it was a moment, I'm like, wow. I didn't know that one day I would be... I had been in such a powerful room. You know, so yeah, that was an highlight. And I had never met... Apart from our first ladies, you know, so... That was an highlight. That was an highlight. Then maybe I also met with my Future X this week. It's just that you don't know her, but that was also an highlight. But how is she your Future X? Tabiazake. Na iso tabiazake. Uka jo weo ni Future X wapo. Awe ni sasa lakini kama mnawistia na time ju sasa kama usha onna. This is a boyfriend. Una ju weo ni Future X waku wukichiaza kidogo. Okita mnuchiazo. Okileta mnuchiazo. Okichiaza mnuchiazo. It has been confirmed. They are no longer together. Shetanya, Shinre. Chupukizi. Thank you so much. Chupukizi, thank you so much. We can't wait to see what you do on stage today. We are cheering you on. Keep going, keep entertaining us. God bless you, you're looking the path, by the way. Santi, thank you so much. Alright guys, thank you so much. That was a comedian Chupukizi who is recording his first comedy show today that is going to be going live on KBC every Monday from 8 to 9. Thank you so much for watching Bazzits Taz. My good name is Miss Kethenshi.


This is the following statement that suppose SL2R, again, acts on a probability space and the action is ergodic, then the action is actually mixing, which of course implies everything that we spoke about yesterday, but well, usually it's the other way around that things are built. The proof is goes along, goes as, I mean uses Moutner phenomena and some other facts. And let me just recall what mixing means, mixing, if you've seen the definition fine, if you haven't, this is a definition, for any two functions f1 and f2 in L2 of x mu, if you look at the inner product, gn f1 f2, this inner product converges to integral of f1 with respect to mu, integral of f2 with respect to mu, for any sequence n, for any sequence gn, which goes to infinity in SL2R. SL2R has a topology because it's two by two matrices, for any gn such that gn goes to infinity. And this is the inner product with respect to mu, of course. Statement clear? And I'm not going to give the proof, but I just want to maybe mention the steps. So one thing that this is an exercise that you should do and it was buried in the last part of second exercise yesterday, that SL2R can be written as SO2, diagonal SO2, the carton decomposition. And well, I think actually in the third set of exercises, this for gl and qp was mentioned. This is a simpler one. And well, the proof is that you look at g, you look at gg transpose. That's a positive matrix. You can diagonalize it. So that's basically what it is. And this one goes by the name of carton decomposition. After this, you can write gn as an element in SO2, a diagonal element, another element in SO2. Compact things cannot hurt you, so you can reduce to gn being diagonal and use an idea like Mountner phenomenon that we spoke about yesterday. But you need to combine it with one other piece of technology. And what you need is a Lagos theorem, together with compactness of the unit ball with respect to weak start apology. With respect to weak start apology. And if you put these together, yeah, what? You can tell me. Both are three dimensional, right? SO2 is the circle. And the diagonal group is a dimension one. And another dimension is three dimensional. Three dimensional. And SL2 also is three dimensional. And one plus one plus one equals three. Yeah, yeah. As I said, the proof is you look at g, you look at gg transpose. This is a positive matrix. You can diagonalize it and that gives you k1. And the square root of the diagonal matrix gives you a. And then you can read out what a2 is. That's the proof. So that's all nice. Let's leave them though. So now I'm going to maybe finally introduce one homogeneous space. We actually did introduce a homogeneous space yesterday, SL3, R1, SL3, Z. But we didn't prove that this was finite value. So I hope to give a proof of maybe in dimension two, a proof that you can then generalize it to dimensions higher using induction. So, well, as always, we're going to start with the general setting. But in a minute, g will be SL2. So g is a locally compact second countable group. Examples are n, zn, all discrete countable groups or SLNR. Gamma subgroup is called a lattice. Gamma is discrete. And second, g mod gamma has a finite g invariant measure. So first of all, these assumptions here tell me that I have left g invariant measure, the R measure, which is unique up to scalar. And a lattice is a discrete subgroup, which is kind of everywhere in your group. So when you take the quotient, the quotient is a finite measure space. It doesn't have to be compact. And actually the example that we are going to be working with, this is not a compact space. But it carries a rational finite measure. So what are the examples? The example is this and this inside Rn. This is a lattice. Why? Because Rn mod zn is the n-dimensional torus, and it's compact. It's a compact group. It carries an invariant measure, namely the Lebesgue measure. If you push it forward to tn, you get a measure. That's an easy example. Still easy, but maybe let's clear. If you take the group of strictly upper triangular matrices, this is not an abelian group. If you choose the entries to come from z, and you choose the entries to come from R, this is a discrete group, and the quotient is again a compact manifold. It's a nil manifold. This is an important group, and this is a nil manifold. And the example that we are going to be mainly concerned with is discrete subgroups of SL2R. So gamma equals SL2z, or gamma equals fundamental group of a surface of genus G with G bigger than or equal to 2.  They are subgroups of SL2R. They are discrete, and they are lattices.  This is z that is R, starting from my hand, rating this is. I'm going to give a proof that generalizes to higher dimensions, and this proof using the action on upper half space, well, doesn't exactly generalize. So remember from your function analysis, well, from your complex analysis course, that SL2R acts by Mobius Transformations on H2, where H2 is the pair xy, y bigger than 0, and it preserves a metric. The metric is, I need to give you an inner product, and the inner product is this. This is the metric it preserves. This is the metric of constant negative curvature, and any surface of genus bigger than or equal to 2 has universal cover, which is the unit this with respect to a corresponding metric, or H2 with respect to this metric. So fundamental group acts on H2. It's the group of depth transformations. It's discrete, and the surface being compact exactly tells me that I have a compact set that I can put it everywhere with elements of gamma and cover the space. That means H2 modulo gamma, if gamma is pi1 of SG, this is compact. It's your manifold. Now what does H2 have to do with my SL2? If you look at this, and let z equal i, then you see that stabilizer of i in this action is precisely the group SO2, a compact group. So this is nothing but SL2R mod out by SO2 mod out by gamma. That was compact, and this is just a compact bundle over a compact space. It better be compact. So this guy is going to be compact. That is showing this one is a lattice. Okay, this was fast, but it's okay. We are not going to use it. You just take it as a black box. That is a lattice. Maybe I should draw the fundamental domain for the action of SL2Z in H2, and then we will prove differently that this is a lattice. This one you have also seen in any sort of complex analysis course that you take. You talk that SL2Z, the action of SL2Z takes this and ties the upper half space. So that is a fundamental domain for the action of SL2Z, and probably you have checked it using the following fact that this one acting on Z sends Z to Z plus 1. So I can translate this right and left, and this element of SL2Z acting on Z sends this to, it's an inversion, so I can bring that downwards, and you can start tiling. That's how one can show SL2Z. This is a fundamental domain, and hence it's a lattice. Well, why is this a lattice? That there, even though I didn't do things slowly and precisely, at least I could hand wave and say this is compact, so we are happy. This is not compact. It's unbounded. It goes up. Why does it have finite measure? Because the measure that SL2R leaves invariant is the following measure, and now you start integrating. The integral is between negative one half, one half, and for every X I need to integrate from that to infinity dx dy over y squared, and hopefully everybody can show that this is a finite area. It equals this one I know, equals pi over 3. So that's another way of seeing SL2Z as a lattice, but as I said, forget about all of that, here's a proof that SL2Z is a lattice in SL2R. I'm going to give another one. I will give another one. I'll give another one that does not really care much that this was two. And, well, the reason I do it is I want to identify this space of lattice. Oh, come clear, why I'm wasting mine and your time doing it. It's okay. So yesterday I said that SLNR, modular SLNZ, this can be naturally identified with the space of discrete subgroups of RN. Now we have a name for them, lattices in RN with volume of RN, modular lambda equals one. So in order to show this thing as finite volume, I need to be able to find good basis for lambda. So you give me a lattice in RN and I want to find some sort of reduced basis. Of course, even for Z2, okay, if you ask the same person, they would choose E1 and E2 as a basis, but if you ask a crazy person, they might choose different basis because you have many vectors that they will generate Z2 as a lattice. Now, what I want to do is to find the basis E1, E2 for Z2 or for any lambda that you give me, what is somehow a reduced basis that I can find for it. And I'm going to do this in group coordinates because then I can give you the formula for our measure on SLNR and tell you to integrate just as I integrated over there. So I'm going to do it in group coordinate. Yeah, you wanted to make it work. So what are the steps? Step one, you give me a lattice. This is a discrete subgroup. I can look at all its vectors and find the shortest vector. I can always do that because it's a discrete set. So given a lattice and now I'll work with two. And of course, you can guess what the exercise is. So G times SL2Z, then GZ2 is the lattice. So this equals GZ2 but I don't really want to look at that. I want to look at G gamma E1. These are not all the vectors but they are the primitive vectors. But if I'm looking for shortest vector, I better look amongst the primitive vectors because if a vector is twice the other one, the one that was smaller is I mean the one that's twice this one, this vector is smaller. So I look at G gamma E1. This is G applied to the set of primitive vectors in Z2. There exists some gamma. So replacing G by G gamma is less than GV for all V in Z2. Is it clear what I'm even trying to do? So what am I trying to do? I am trying to find some little gamma in SL2Z such that G gamma has a simple description. Maybe you gave me G but in this coset space, there is no unique choice of G. I can replace G by any G gamma and it's the same point in the space. So what I'm trying to do is maybe you gave me G that the coordinates, the 2 by 2 matrix that represented G was very tilted. Some coordinate was huge, some coordinate was small. I'm trying to find some integral matrix so that if I multiply G by that, this has a very nice form. What kind of form? In a minute. That's what I'm trying to do. So I'm trying to replace G by G gamma so that this has a very particular form. And that's how this reduced basis is found. Step one is that I can replace G by G gamma and assume that this holds. This already puts quite huge restriction on G. Why? Because of first an exercise. This is usually referred to as the compositional. SL2R can be written as SO2 times A, the diagonal, times the unipotent that we spoke about yesterday. Okay? Okay, I gave it a nice name. Of course, it goes with this name but there's a different name for it. What's the name that we all know? It's the Gram-Schmidt process. You give me a basis and if you Gram-Schmidt it, this is exactly what you get. So the composition that Gram-Schmidt process gives you. So acting by elementary matrices you can make them orthogonal, then you have the norms and then you get orthonormal. That's what it is. So I have this decomposition. Now the claim is that if this holds, then if I write G as KAU, A is very specific. Under star, if we write G equals KAU, then it equals e to the t, e to the minus t and e to the 2t is less than 2 over root 3. Here also this is less than or equal. There may be that few vectors have the same norm. How do I get this? I know that some vector will have minimal norm. This vector better be primitive because I can always divide and make it primitive. The primitive guy is going to have smaller norm. Primitive guys are orbits under e1 with SL2Z. I replace G by the corresponding element and just assume that the vector was e1 to begin with. Does that make sense? So the most important fact that was used is that this is a discrete set. So it takes its value, its minimum and then I just call this new vector e1 by moving my G. Now I stop here and say we are done. If I show this, we are done. Why are we done? Let me first say why we are done and then maybe I actually leave this to you as an exercise to prove. Why are we done? Because u is harmless. u looks like that. I can replace this by a translate of it by integer so I can add n where n is in integers and make sure this is less than 1 half. This I can always do. I am free to move by elements in SL2Z and if I do that, this is a matrix multiplication even I can do, I am going to get that and then choosing n, I can guarantee that this is less than 1 half. There exists some n in z such that that happens. So what did I show under this star start? So we showed there exists some gamma in gamma in SL2Z such that this gamma is different from that gamma perhaps but such that g gamma can be written as something in SL2, some diagonal element with e to the 2t less than 2 over root 3. This particular number is not very important. Any number would do but this is the optimal times 11x where x is less than 1 half. I am going to replace this by big k. I have a compact set, a compact set and I said that is not compact. This looks much like that. It is a compact thing but there is one from here to here is a compact direction. I go up and the k was not there because I modded it out but it is not exactly this. So I found a representative in this set. This set has a name. It is the Ziegels set with 2 over root 3 and 1 half and the claim is that this has finite value. Of course 2 over root 3 comes from here and the 1 half comes from there. So it is all the orthogonal matrices times the diagonal guys with this restriction times the unipotent guys with x less than 1 half. I claimed that SL2z was a lattice so I better be able to show this thing has finite volume. This is again another exercise. Recall that the harm measure for SL2r can be written in this coordinate as this is the rotation group so there is d theta then I have the diagonal group dT and of course I have used the notation I used T too many times dS except there is a weight and the weight is 8 to the 2T. This is the harm measure where I decompose G as cosine theta sine theta minus sine theta cosine theta e to the t e to the minus t and 1 1 s. Now calculus exercise is that this region has finite area with respect to that measure. Volume of S2 over root 3 1 half is finite and this shows that it is a lattice. Another exercise which is not a calculus exercise is that prove SLNz is a lattice in SLNr. The idea is very similar you need to do induction and it's a good exercise. Any questions? Yeah and one unbounded direction. It's integrability exactly because if you integrate with respect to that it becomes integral from negative infinity to whatever e to the 2T dT integrable function. Okay so now why did I do this? It's a good question. I want to now state Mahler's compactness criteria. When do I stop? Let's say 20. Yeah let's say 20. So I have 20 minutes. Okay so what is Mahler's compactness criteria? It's SL2z, SL2R model SL2z which now I'm going to give it a name X sub 2 so X sub n is the space of lattices in Rn the space of unimodular lattices in Rn meaning that the volume of the quotient equals 1 so this is what Xn is going to be. For any epsilon I'm going to define Xn of epsilon or X2 of epsilon to be the set of lattices. Lambda is a lattice in Rn, covalent equals 1 such that the infimum of normal v we just said this is a minimum actually so maybe I'll write minimum v in lambda non-zero is bigger than or equal to epsilon. So it's the set of lattices that they don't have short vectors. A corollary of what we just did is that X2 epsilon is compact for all epsilon positive. And of course they exhaust the space. The space can be written as union of these but this is a compact set. Why? Because let's go back here. What does that condition the infimum of non-zero vector bigger than epsilon do? I have an upper bound. It puts a lower bound here. So it just cuts this. It gives you a compact set. Now why do I care? This is a non-compact space so when you have a flow on it it's not even clear that this flow is not divergent. It doesn't go out to infinity. Of course we have point carrier currents which tells you almost every point for a flow is going to come back but given a particular point this is not at all clear. So the action of the group U is a very particular action and you have a non-divergence. Before going there let me give an example of not non-divergence. Consider T going from X2 to X2. T is multiplication by the matrix 2 1 half. I have the space any subgroup acts from left so I can let this one act. Concareer currents tells me that every point will come back almost every point will come back close to itself infinitely often. But it doesn't tell me what happens if I give you a particular point. So I'm going to give you a particular point namely the standard lattice. And Tn of X is divergent. It leaves every compact set. Why is that? What does it mean to leave every compact set? Let me show you that I can produce lattices with short vector in this orbit. For every n or every epsilon positive there exists some n null such that Tn of X is not in X2 epsilon for all n bigger than n null. That's what it means to leave every compact set because of Mahler's compactness criteria. This has a name. But why is that? See what happens to the vector E2. The vector E2 is contracted because I'm multiplying it by an eigenvalue less than 1. So the vector E2 just goes to 0. What I'm doing is taking this standard torus and taking this curve and just pinching it to 0 along the orbit. So this in the orbit goes to a degenerate torus. So it's not recurrent. It leaves every compact set. The action of U is a completely different game. For position lambda in X2 one of the following calls. Lambda has a horizontal vector or there exists some capital T which depends on lambda such that well maybe this is Tn null. For all T bigger than Tn null if I look at the times in the interval 0T such that my unipotent ut ut remember what it was. It was this guy. Applied to lambda lives in X2 of 1 over 1000. This is a pretty random number. Some epsilon. This measure is bigger than 0.9 times T. So not only do I come back to a fixed compact set I come back a lot. I spend most of my life in a fixed compact set. So this is in strict contracts with this statement that I leave every compact set. This flow leaves every compact set. This flow either something very algebraic happens you have a horizontal vector or you are biased towards the compact set. In dimension 2 this is easy enough that hopefully I will finish the proof today. Dimensions higher this is very nontrivial result due to Margulis which was one of the ingredients in his proof of super rigidity. Not even this but just to have infinite return to a fixed compact set. And this was generalized by Danny and then Climax and Margulis with applications to life and time approximation. Dimension 2 this is really an exercise so we are going to do it. Of course I'm not going to prove that. I'm going to prove something that implies this and to connect these dots is an exercise for you. So the following is the proposition I'm going to prove. For every interval I subset of R the following holds that lambda in x2 and rho be a number that is positive but less than one half and assume that the supremum of norm of utv t in I is bigger than or equal to rho for any non-zero vector. Then for all epsilon that's less than rho over 10. Let me write it and then I'll explain what this means. So it's a mouthful. What is it saying? I have my lattice lambda sitting in R2. It's got a bunch of points and I'm looking at my flow ut and I want to see what happens to this lattice. I am looking for a statement like this that the lattice comes back to compact set but the lattice is not one point it's infinitely many points sitting everywhere. What this proposition is telling you is that look at individual points and suppose well of course individual point think of themselves as center of universe so pretend that's the case just follow what happens to this along the orbit forget about the rest of the lattice. So I just look at what happens to individual points and I put this condition that the orbital points grow so if I look at this and hit it with my unipotent which by the way if you look at the action it's just straight line because this point was x, y you hit it with the unipotent it's x plus ty, y so it's straight line and assume that this is really the picture that your points move and they attain certain supremum so the supremum grows becomes bigger than a row then I can find times that if I look at all of points moving together none of them is in the small blocks that's the statement down here I can find times and lots and lots of them oh sorry so that's probably I can find times and lots and lots of them so that no points from this infinite collection comes to the small blocks that's the statement okay I just corrected there was a typo with that should have been bigger than 1 minus 3 is the statement clear? was it? intersection of what sorry now what's the proof? the proof is relies on dimension 2 really and if you want to generalize this to higher dimension you need to be more goalist and okay, art so first of all I'm going to make my life easy and work with max norm of course the statement the result all of it is irrelevant I mean it's insensitive to the norm because all norms are equivalent to each other so I'm going to work with max norm and I am going to recall or point out the following special property of dimension 2 if lambda in R2 is a unimodular lattice is a lattice with volume equals 1 then you cannot have two linearly independent short vectors period then you can find v1 and v2 linearly independent both in the lattice with norm of them less than a half just can't because if you did then they would give you a fundamental domain and the fundamental domain is going to have volume less than 1 and this is very crucial because I am going to look at the behavior of all points and this will give me some umbrella that if one vector is short shorter than one half no other vector is shorter than one half this is extremely important it's not important but I'm going to work with one half anyway I mean it would be bigger than or equal to so okay so I have the interval I I'm going to cover it with sub intervals and use conditional star to say that this is a disjoint covering and then inside these intervals I am going to control when vectors are short that's the strategy for every v in lambda define IV of rho well we said it's not important but I need a notation so IV of rho to be the set of times in my interval such that the vector v has norm less than rho that's not equal to rho so let me draw this picture here I'm working with max norm and this is the robot I'm looking at all the times where my vector along the orbit comes in the robot okay star implies IV1 does not intersect IV2 so the collection of IVs gives me a covering disjoint covering of a subset of I in particular what does this imply this implies that the measure of union IV sum of the IV rows equals the measure of union and this is less than or equal to measure of I that's what it implies I can take the sum and the measure does not go up now inside each of these intervals if I have short vectors what does it mean to be in here that means some vector became short inside each of these intervals I'm going to detect the bad intervals what are the bad intervals the bad intervals are this vector being less than rho does not hurt me it hurts me when it's less than epsilon and I'm going to control those times these in the vector being IV rho because of what I'm about to write is less than epsilon okay so now I'm going to look at within this interval that this vector the orbit of this vector entered the umbrella I'm going to look at whether or not it hits the dangerous box and I would like to show that the measure of this is less than the measure of that with actually a definite factor then I'm done because I know that happens each of these have measured less than that and the bad times are contained in the union of these that's what I need to show so let me write a claim oops this was the whole proof sorry so what is the claim if IV rho IV epsilon sorry is not empty suppose for some V I have a time that I become less than epsilon then the measure of IV epsilon is less than epsilon over maybe 2 epsilon over rho minus epsilon times the measure of IV rho I'm going to be down special I can't leave it like this I need to finish so because there was complaint this is exercise this is where you use this condition and let me draw the proof of it if you're here that means that the speed see the speed is given by Y if you just look at what it means for a vector to have that property and because you need to get out of this the distance because you're supposed to reach the boundary then you will have at least one of the end points for getting to rho will have to be contained in I that's somehow if it's not clear ask me later okay now finish the proof this set Utlanda is not in X2 epsilon this is contained actually equals to union of IV epsilon by definition now the measure of this is less than or equal to the sum of measure each of them and this is less than or equal to that 3 epsilon over rho and this is less than or equal to 3 epsilon over rho times measure of I by what I wrote over here if you prove the claim then you're done and the claim is the fact that linear function this was actually quite an important linear functions if they are small for time it takes them quite some time to grow that's what it is so I'll finish with this exercise which is a generalization of the fact about linear functions suppose F is a polynomial of degree D or whatever then the measure of times so that F of T is less than epsilon times the supremum of F over I this measure is less than C epsilon to the alpha times the interval there exists C and alpha depending only on the degree such that this happens and this is a generalization of in here you're just really dealing with linear functions but that's okay, sorry


Big moment here, early on at the King Tower, it will be Billing! And it's only just wide, so, so close to an opening goal. Off-well, swings it in, Lerma was trying to get on it, he'll break to Billing, who just held his position on the outside of the box and he tests Stanya Leverson there. Second time that Bill Billing's gone close this afternoon. The ball to find Uttara, who finds Solanki, who, on the turn, puts it the wrong side of the post. Well, risky ball back here and Billing's in for Bournemouth! It's a gift of a goal! And it's a goal that has been coming. But yet again, Leicester City shoot themselves in the foot and on this day of all days, that is an almighty mistake to make Bournemouth with the breakthrough. And you have to say, it is thoroughly deserved. Able to break in behind the Leicester defence, it looks to link up with Solanki, who takes the touch! And it's a telling and vital intervention. Bournemouth threatening so early in this second half, whipped in by Rothwell. Towards the back post, Leverson keeps it out the first time. Christie heads it towards goal and Leverson gets his hands on it. Bournemouth really, perhaps, should be further in front, Leicester are hanging on in this game. Estanya to Madison. Tictoes his way into the Bournemouth box and it's tipped over the top by Neto.


Opportunity for Lewis Cook, and now it's with Jack Stacey. 30 yards out, he's going to size up a shot. Stacey, he shoots! Oh, Jack Stacey opens his form with a magnificent strike from 25 yards off the inside of the right-hand post. And the Jason Tindall era is up and running thanks to Jack Stacey with a Thunderbolt. That's been given away by Blackburn. They've got a man committed upfield now. And here's Adam Smith, leading the charge. VanJuemma left corner of the penalty era. Arnaud VanJuemma in the box onto his right foot. Magnificent from the Cherries pre-season in for man. And the Dutchman has his first competitive Bournemouth goal.  Stacey wins it. He was hungrier than McLean. And now Lerma with bidding in space to his left. Oh, what a turn from Dan Duemma at the edge of the era. Can he finish? Dan Duemma! To go until half time. Bournemouth have it edged in front against Norwich City. 1-0. The goal scored 4-0 at Bournemouth, Arnaud Dan Duemma.


Hey, welcome to this week's session where we're going to discuss introduction to basic probability. I know that in your module, you don't deal with probabilities in that extent, but there are a couple of topics or a couple of concepts that you need to be aware of and a couple of calculations that you need to be able to do as well. So we're going to try and see if we can get you to that point where you are able to read a probability question and be able to answer that. So by the end of the session today, you should be able to learn the key concepts of probabilities, when things are events, the types of events that are there, like your mutually exclusive events, your compliment events, your independent events, your simple and joint events, and so on, but also more specifically in terms of calculation, you need to know when to apply the multiplicative rule and when to apply the addition rule as well. Okay, so in terms of probabilities, we know that the probability is a term that is used for likelihood of events happening. So a probability can also be described as a chance that a certain event will occur and it will always be between zero and one. So a probability, it's a numeric value and you can represent it in a decimal format or in a percentage format. So in a decimal format, it's between zero and one, so it can never be a value in a negative or it can never be a value bigger than one. If we represent it as a percentage, it's also it's zero and 100%. So it can never be any value in the negative percentage and it can never be any value greater than 100%. When an event is certain to happen, for example, the sun coming up, that event, we call it a certain event and it's always going to have a probability of one. An uncertain event or an impossible event is that impossible, is that event that has no chance of happening and that always have a probability of zero. Now, when we assess probabilities as well, there are three different types of probabilities, the three different ways that we can assess probabilities. We can use the priority, which is the likelihood of an event occurring when there is a finite amount of outcomes that exist and we always use this. In most cases, the probabilities that you're going to be working with in this module are always the priority, they always use the priority approach and the probability that something will happen in priority, it's the number satisfying that event or that event divided by the total number of outcomes and we also have an empirical probability, which refers to a likelihood of an event occurring based on historical data. Also, the probability of occurrence for that empirical probability is the number satisfying that event divided by the total number of outcomes and both the priority and the empirical probability are also what we call classical probabilities because these are most often those probabilities, the way we approach probability concepts or the way we do probabilities, we use either the priority or empirical probabilities and then the third one is the subjective probability. This one is based on the researcher's opinions. It's a combination of individual past experience, personal opinion, analysis, and particular situation. So this one you use mixed method type of a way of assessing your probabilities because that you also have to put yourself in it, your opinion, your prior knowledge, or the way you view things. Okay, so we spoke about number satisfying an event and the total outcome and so on. What are these events? So for example, an event will be a process of something happening. That is an event. You create an event. So each event has a possible outcome that comes out of it. For example, if I toss a coin, a coin has two outcomes that it can land on. It can either land on a head or it can land on a tail. If I roll a die, which I'm creating an event of rolling a die, and a die can will have the outcome one, two, three, four, five, and six because it's got six sites. Those are what we call possible events. And we can define each event in terms of what kind of an event is that. Is it happening on its own or is it happening and it has other things around it as well, happening with others. For example, we do have what we call a simple event. It's just one event. A simple event describe a single characteristic which that single, so that one icon. So a single event will be the probability, though, sorry, an event that a coin will land on a head. That is one event. Landing on a head is one. Or all students registered in a particular module. That's single event. We also have joint events where things happen together. So for example, when you have two characteristics that define that. So a person who registered in a psych 3704 and is a male or a female. Those are two events coming from two different characteristics. One is the module and the other is the gender. Both of them happening at the same time. Those we call them joint event and they are represented by an end to combine the two events. Then we also have what we call a compliment event and a compliment event will be the event that is included, but not in the original event that you are looking at. For example, if I'm taking an example of a coin and I say I'm creating this event of tossing the coin and it must land on a head. A head is the probability is the event that I'm looking at. The compliment of the head would be the other outcome that is also included, which is the take. So a head, the compliment of a head is a take. A compliment of a take is a hate. And we always denote a compliment by using let us like a compliment with a copy or a copy or a compliment or not there must be subscript, superscript, a compliment and so on. You can denote your compliment event as such. Okay, so when we have a total of all collection of all outcomes, all those outcomes are called what we call a sample space. So for example, all sides of a dice create what we call a sample space. Those are the total outcomes. A coin, total outcome will be all the sides of that coin. Total outcome will be all number of students registered for psych. That will be the total outcome plus other characteristics that they have, but it will be the total outcome will be made up of all students registered for modules in a program. Let's say it's become degree or a BA degree in industrial psychology. Let's say that is the cost. So all students registered for that program, they do different modules. So we need all of them, all those modules, they when we combine all of them, they create what we call a sample space. Okay, so we can visualize probabilities and events or mostly events. We can visualize them to make sense of them as they happen. And we can use what we call a VIN diagram. And with a VIN diagram, it creates a joint event, single event of all possible outcomes that we have. Or we can visualize them in terms of a decision tree which branches off. It starts with the whole or the total outcomes and then you start branching out to say if I look at how many students are doing psych or other subjects. And then those who are doing psych, we can look at their agenda and look at whether how many are doing, how many are female, how many are male. And from those ones, we can also look at other characteristics, how many are in terms of race, how many are black, color, white, Indian, and so on. So you can start branching it out in terms of that to look at the possible outcome. So if we look at the coin, a coin has two sides. So if I toss the coin, the first coin, the coin will land on a head and a tail. If I toss that coin again, if it first landed on a head and now it can either, when I toss it the second time, it can either land on a head or a tail. And you can continue with that to create different scenarios as well. And you can use this to create your scenarios and answer the probability questions based on that. And we can use some of these examples when we look at the exercises as well. You can also use a contingency table, which makes it easy for you to visualize events, especially where you have joint events and simple events and you want to make use of a contingency table. So there's nothing stopping you from using also a decision tree. But for me, contingency table, it's much easier to use and interpret. Okay. So from a simple event, we can calculate a simple probability by using observation satisfying that event of a simple event divided by the sample space or the total outcome. So let's say, for example, this is our data of events happening. So we've got gender and the module register. So if I want to calculate the probability that a student is registered in an STA 1610, then regardless of what kind of agenda they are, I'm using simple evented means I'm only looking at the module. There are eight of them. And the total outcome is the sum of all the other values that are there, which is your sample space. There are 20 outcomes. So I'll take eight divided by 20, and I get 0,4. If I need to calculate the probability of a main because the students are registered in different modules, I'm disregarding that I'm only interested in the probability of a simple event main, regardless of which module they are in, the total outcome satisfying that simple event, they are five. And I divide them by 20 to get my simple event, probability of my simple event main, and you can do it for any other question that you want to answer, especially if you're only interested in simple events. So you need to be able to recognize that this question is asking you to calculate a simple event or it asking you to calculate the joint event. So in terms of joint events, these are events that happen at the same time. So it comes from to calculate the joint probability, you will calculate it from a joint event. So two events happening at the same time, two or more, it can be more, actually. So that is also given by number satisfying that joint event divided by the sample space or the grand total or the total outcome. Looking at the same example, so when do we see joint events, STA 1610 and female, there are seven of them. So it means students registered for STA 1610 and there are female, there are seven of them. Those are joint events. So this is where you calculate the joint event. So let's see if we calculate a probability of a male and statistic or STA 1610. So a male STA 1610 event satisfying is one, there is only one male doing STA 1610 and out of 20 people that are registered for this program. And we say one divide by 20 and that gives us the probability and that is the joint event. Marginal events, I'm not going to touch on the marginal events, it's just the addition of the joint events to create a simple event as well. So I'm not going to touch on that. Then we also have what we call mutually exclusive events. And mutually exclusive events are events that happens simultaneously or at this, sorry. Mutually exclusive events are events that cannot happen at the same time. They cannot exist at the same time. So it means usually this refers to joint events. And we know that the probability of a joint event is number satisfying that event divided by the sample space. But for mutually exclusive event, for example, if I need to choose a random student registered for a program and I choose a student from site 3704 and student from QMI 1501. If on that dataset, all of them, it's individual students, they don't repeat courses. They all do different courses. So student A is from site 3704, student in QMI 1501. Both of them cannot happen at the same time because you cannot be in site and be also in QMI. At the same time, you will be in one. So these are what we call the mutually exclusive event and the probability of A and B. And you will notice that I use not an end end, but I use the intersection end, or you can use the end. It means the same thing, and B. So if I need to calculate this probability, the probability of A and B, that probability will be equal to zero. That is mutually exclusive event. Collectively exhaustive event. These are events that complete the sample space. One of the events must be okay, but they also need to cover the entire sample space.  So for them to be mutually exclusive, head and a tail needs to happen some way, some amount in that. So let's assume that we randomly choose a day in 2014, a day A representing all the weekdays, B representing days that falls on a weekend, and C representing days that falls in January, and D represents days that falls in spring. Weekday and weekend and January and spring, they all create a collectively exhaustive event because they all include all the days of the calendar. But they are not mutually exclusive and weekday can also be in January and can also be in spring and a weekend as well. So if you think about it, collectively exhaustive means all of them together, right? They complete the sample space, but they can also still not be mutually exclusive because mutually exclusive events are events that cannot happen at the same time. So A and B are mutually exclusive because a day in weekday cannot be a day on a weekend, right? So A and B on their own as well, they are also collectively exhaustive. So what have we done so far? Up to now, we learned that we can calculate from using a contingency table, we can calculate the joint probabilities and we can calculate the simple probabilities. And so far what we learned was the probability, it's a numerical value, and it measures the likelihood of an event happening, right? And we learned that the probability has a value between zero and one, inclusive because it can either be zero for an impossible event or one for a certain event. The sum now, you need to also know and remember that the sum of all events or the sum of all probabilities should always be equals to one. And if and only if A and B are mutually exclusive and collectively exhaustive, then the sum of all of them would always be equals to one. The other thing we also learned is the complement event A is defined to be an event consisting of all possible points that are not included in that original event A. So we can use the formula. If we want to find the probability of A and we want to find the complement of it, we can use one minus the probability of the complement or we can find the complement event, the probability of A complement will be given by one minus the probability of that event. They will all give you the same. So if I've got a complement, I can calculate the probability of the event or if I have the probability of an event, I can calculate the complement and going to learn all those things. Then we also have what we call probability rules, and one of them is calculating the event that either or the two of them happening at the same time. So if I need to calculate the probability of A or B, which can also be denoted by the probability of A union B, you can either use the two. It's given by the probability of the simple probability of A plus the probability of B minus the joint probability of A and B. Only if and only, if A and B are mutually exclusive, then the probability of A and B is equals to zero because we've learned that the probability of A and B, the joint probability of A and B, if it's equals to zero, therefore it means the events are mutually exclusive. So only and only if A and B are mutually exclusive, then the addition rule states that the probability of A or B, the probability of either A or event B will be given by the probability of A plus the probability of B. That is only if they are mutually exclusive. We also have what we call conditional probabilities, and these are probabilities of one event given that another event has occurred. And that is given by the probability of an event A given that the probability of B has already happened, is given by the joint probability of A and B divided by the probability of B. And you can write it vis-a-vis-a as well. When events are independent, then the conditional probability states that because of the two events being independent of each other, it means one event does not have any influence on the other one happening. So when one happens, it does not affect how the other one is happening. So two events are independent, if and only if the probability of a conditional probability of A given B will be given by the probability of A because B has no effect on what A is gonna be. And we can do the same. So this one is a repeat of that. Event A and B are independent when the probability of one event is not affected by the fact that the other event has already occurred. And that is independent. So then how do we then find the joint probability of A and B if we've got independent events? So, and that's when we introduce what we call a multiplicative rule or multiplicational rule for two events. So when two events are independent, we know that the probability of A given B from the conditional probability, let's put it this way. We know that the probability of A given B is equals to the probability of A and B divided by the probability of B. This is for normal probabilities. Now, if I'm given the conditional probability and I'm given the simple probability, I can then in a mathematical format, we say we cross multiply. In order for us to remove the probability of B, this site going to multiply by the probability of B, whatever you do on one site, you must also do on the other side. Therefore, this one will cancel out and you will be left with the probability of A and B is equals to, I'm gonna start with this one. First, the probability of A given B times the probability of B as you can see there. That is what we call a multiplicative rule. That's where it comes from. However, if we're dealing with independent events, now if we're dealing with independent events, we learn that the probability of A given B is equals to the probability of A. Now for independent events, and if and only if A and B are independent, then the probability of A and B will be given by the probability of A multiplied by the probability of B because we're going to replace the conditional probability with the probability of A because the conditional probability of A given B for independent events is equals to the probability of A because B has no effect or does not affect the outcome of the probability of A. Okay, so we've learned the multiplicative rules. Now, let's do more examples. But before that, in summary, just to conclude as well. Independent events, remember, is when occurrence of an event has no effect on the probability of the other event happening. Mutually exclusive event is when occurrence of one event precludes the occurrence of another. Or we can also say, mutually exclusive event is when two events can never happen at the same time or cannot happen at the same time. Exhaustive events are a set of all events representing all possible outcomes of a sample space. The additional rule is when events, additional rule occurs when events are mutually exclusive. So you must ask yourself events mutually exclusive and am I calculating an event of either or? Then I'm going to apply the additive rule. The law of conjunction deals with independence. So it meant to go into ask yourself this one, am I dealing with conditional probabilities here? Does event A and B independent from one another? Or are these two events independent from one another? So when they are, then we're going to use, especially for joint event, we're going to use the multiplicative rule because the probability of A is equals to the probability of A. Probability of A and B is equals to the probability of A plus the time, the probability of B. Always remember that. Okay, so let's look at activities. I'm going to do the activities with you. We'll do them together, so. Let's see. In the population, there are 450 people of whom, 150 smoke. What is the probability of randomly selecting a non-smoker? So if this is our sample space N and if this is our X and if non-smoker is X complement, or let's not make it an X way, let's make them A and B. A and non-smoker is A complement. So we need to calculate the probability of an A complement. We are told we can calculate the probability of A which is your X divided by M, which our X is 150 events satisfying, which is the smoking, divided by 450. And that gives us 150 divided by 450, 0,33. 0,33. Therefore, to calculate the probability of A complement, which is non-smoker is one minus the probability of the event. So which is one minus 0,33, which is 0,67, which will be option one. That's one way of doing it. The other way of doing it is, I know that there are 450, 150 are smoking. Therefore, A complement will be 450 minus 150 and that will leave me with 450 minus 150 leaves me with 300. So in order for me to calculate the probability of non-smoker or I can say it's the probability of A complement because I use the A's, I'm gonna use 300 divided by 450 and 300 divided by 450 is equals to 0,67. So you have two ways that you can use based on how we answered the question. So you can either answer it using the probability concepts or you can just go ahead and calculate the event and just do that. Easy, easy. Moving on to question number two. Which of the following does not represent a probability? In your module, I think they expect you to use probability in a decimal format and you will use proportions in a percentage format. So which one does not represent a probability in this instance, it will be option number one, which is in a percentage. So proportion will be percentage probabilities will be decimal. Okay, question number three. If 10,000 students who wrote a university admission test 7,000 passed, it means they obtained 50% or more. So it means they got greater than or equals to 50%. So it includes 50% and 400 only obtained exactly 50%. So out of this 7,400 got 50, so what is the probability of a randomly selected student will fail a test? No, this is just to confuse you. Read the question carefully and make a decision from there and say which statement is relevant to answer this question and which statement is not. So let's start with that. We have our sample space and we have our event A and we also have our joint event. I'm not gonna say it's joint event because this is exactly, I can call them B as of exactly. So B is equals to 5,400 and it is also included in this. So now what is the probability of a randomly selected student would fail? What does fail mean? Fail means the student would have received less than 50%. So if I have greater than or equals to 50%, then I've got my entire of those ones. So the rest means they failed. So 70, 7,000 passed, the rest from 10,000 failed. So our B fail, I'm gonna use this was for this one, and this was for this one, the B there. And I'm gonna use A complement with this one. So A complement is 10,000 minus 7,000 is 3,000. So 3,000 of the students failed. So in order to calculate the probability of a fail, we're going to say 3,000 divided by 10,000. And the answer is, the answer is 0.3. And it's none of those ones that are there because they're looking for the probability of those who have failed the test and to fail the test, they've received less than 50. Unless if they say they obtained 50% or more would mean it does not include the 50. Let me just see if I don't include from the 3,000. If I subtract also the 400 to get to 26,000 divided by 10,000 but this is very wrong, it's 26% so it's not. So I think one of the options here is written wrong. Maybe probably I'm gonna assume that this one was 0.3. Okay, let's look at exercise four. A probability of an event occurring which depends on something else occurring such as passing a test when you do not understand your course can be described as what is the probability of an event happening given that another event has happened. Is it one conditional probability to independent events, the real mutually exclusive event or multiplicative? Independent event describes the event here. We're talking about the probability. Mutually exclusive event describes the event. It's not a probability multiplicative probability. We've never spoken about such thing. We know that this should be multiplicative law, not probability so the only one that is outstanding is just the conditional probability and we remember conditional probability is probability given that another event has happened. The formula, we write it that way, right? And that is the definition of a conditional probability. Let's look at exercise five. Yo, you must let me know if I'm going too fast. Let me just ask you to answer this one so that then it doesn't mean like I'm giving you all the answers. A ball is drawn at random from a box containing six balls. White, four, six red balls. Four white balls, five blue balls. What is the probability that it is red? So remember to always calculate your sample space N and your sample space N will be a total of all outcomes. All outcomes is all of the balls. So there are six plus four plus five, which is equals to 15, that is your sample space. So calculate the probability of red. That will be outcome satisfying red divide by N.  Divide by the total sample space. What is the answer? Are you still there? So the answer would be observation satisfying red. They have six of them. The sample space is 15. And what is the answer? Six divide by 15 is 0.4 and we can just add the zero zeros because they put your answers in three decimal. You can just add the other zeros because it doesn't change the number. Let's look at another one. The probability that Jamie will pass his research methodology exam is 0.5. Find the probability that they will fail. The probability of fail is given by one minus the probability of pass. So they have given you the probability on red. So they told you that this is a probability. Always remember, one, two, three, 100, 16, 20, 10,000. These are what we call events. If it's 0.1, 0.30, 0.5, 0.78, these are what we call probabilities with an exception of one. So one can either be an event and can also be a probability. Always remember, if they give you the event, you calculate the probability. If they give you the probability, you just substitute. So this will be one minus 0.53. And the answer is 0.47, 0.47, which is option four. Which is option four. Okay, now we go into interesting stuff. If a coin is flipped three times, the sample space of all possible outcomes would be. So now, in this instance, you can use a decision tree because they say a coin is flipped three times. So we're going to create three, three ways. So we start at the beginning and we branch out. So it was a head and a tail. And at that point, we taught this coin. So this is the first time, right? The second time, it creates a head. Let me write on there. I'm gonna do it like that.  So those will be our three.   A head and a tail, even though they are not that visible. So this one is head, head, head. Head, head, head. This one, it's head, head, tail. Head, head, tail. And this one and that one, I'm gonna do all of them like that. I'm gonna have all the outcomes. So this one is head, tail and a head. Head, tail and a head. And this one is head, tail, tail, head, tail, tail. And this one is tail. Oh, I forgot to put head and tail on this.  And this is a tail, head, tail. Tail, head, tail. And this one is tail, tail, head. Tail, tail, head. And this is tail, tail, tail. Tail, tail, tail. And those are the three possible outcomes. So let's see. Head, head, head. Head, head, tail.  Head, tail, head, head, tail, head. Head, tail, tail, head, tail, tail.  Tail, head, tail, tail, head, tail. Tail, tail, head, tail, tail, head. And tail, tail, tail, tail, tail. You see how easy it is to find your outcomes. If I look at the other ones, they could also be, but they might be repeating themselves. Head, head, head, head, tail, tail, tail, tail, tail, tail, tail, tail, tail, tail, tail, tail. It's repeating. This one is repeating. So it cannot be that one. So the others, they are missing other things. So you can see. So let's do the last one, or this is the second last. Then you flipped four unbalanced or well-balanced coin. What are the odds that two of the coins will lead, will lend heads up? Also, you can create a decision tree on this one so that then it helps. So it means I have to draw a big one now. So we start at the beginning and it lands on a head and a tail and going to split two. That's one, two, three, and this will be the last one. It should do the same with this one. It has to split twice and I'm going to run out of space. It has to split twice and it also has to split again. And the same on this one. It has to split and it has to split and it has to split again. Okay, so we always start with the head. So it's head, head, tail, and on this one, head, tail, and on this one, head, tail, and we do the same.   Okay, so I've created all those scenarios. So I'm doing it this way so that I know that I don't skip anyone. I include, I'm going to run out of space. Yeah, so the first one is head, head, head, head, that line. Head, head, head, head. This one, head, head, head, tail. Head, head, head, tail. The next one is head, head, tail, head, head, head, tail, head. The next one, head, head, tail, tail. The next one, head, tail, didn't put the head, tail. The next one is head, tail, head, head. The next one, head, tail, head, tail, head, tail, head, tail. The next one, head, tail, tail, head.  Tail, sorry. That's another tail. The next one, head, tail, head, head, tail. Tail, head, head, head, tail, head, tail.  So I've got all scenarios. Now, what are the odds of that two coins will land on a head? So we need to find where there are two heads. Let's change the paint color. I know that we, oh, sorry, that is the last one. I just want to go in here and change my pointer color. So we want to find where we've got only two heads in there. One, head, head, head, there are three, two, two, there are three, two, two, two, and those are the ones. So one, two, three, four, five, six, six, two heads coin. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, is our total outcome. So six divided by 16, which is the probability. If we're going to call this A, I always label it my events A, A, A, which is 0, 3, 7, 5, which is option three. That's how easy it is, or how tricky it can be to answer questions on probabilities. And that concludes today's session. 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Enjoy the rest of your Saturday. Thank you. You too. Bye.