Afternoon Gary. Hi Mark. Obviously last week was a real blow for you and the lads and the club having played so well at times at the Emirates. How much of a psychological coaching session have you had to do with the lads to pick them up? Because conceding in that manner, in that style, when there was so much at stake, clearly the lads would have been disappointed. Have you had to kind of lift them in a psychological way? I think the moment in the dressing room right after it was the toughest aspect. So speaking to the group about how I felt about the game at that moment, understanding where they were emotionally from that. But moving past that this week has just gone back to Liverpool coming. So from Monday the lads were ready to work, ready to go. So there was absolutely no hangover whatsoever. There was a review of course of things we could have done better and how well we did to go there and make it as close as it was and be in the game for 97 minutes or whatever we were. But yeah, so that in the dressing room making sure we make sure we know what happened out there.  Not many teams, I don't think many teams at all have gone there and scored twice and been ahead or level for 96 minutes at the Emirates this season. It was a big effort from us and then of course the disappointment at not getting anything from that. But yeah, the lads are fine this week. We're ready to go. Another big, big side coming. So full focus from Monday onwards on Liverpool. At one stage on Saturday you were fifth bottom in the Premier League. You go to bed bottom of the table. What does that say about the nature of the league at the moment and how defining every single moment can be in a game to where you end up finishing? Yeah, the margins are so small as you guys already know. So yeah, the nature of every chat we have after every Saturday game with you guys changes in an instance with real fine margins. I think a lot of the last few games, including the one we won by the way, including Wolves, fine margins at Wolves as well. So they can go in your favour. The gold, not in a forest score here where Sam Surridge last year would have been technically offside by last year's rules. So the margins are tiny and we are at one point behind achieving what would be a real successful season for us at the moment to stay in the Premier League would be a huge success. We're one point behind that at this moment. There are six points that separate the bottom nine. For you as someone bottom looking up, does it feel like there's nine teams in this relegation scrap? And the fact that there are so many sides that are in amongst it, does that give you a belief and a confidence that you can finish outside of the bottom three? Yeah, I think it's a difficult season to sort of pick the ones that you expect to be right down there. Come the end, I think there is a bit of a mixed group there at the moment. And of course when you're in and amongst it, the more teams that are involved, the better. But yeah, it is a very, very tight bottom in the league. So back-to-back results or a win that people don't expect you to get can make a big difference. Just a quick word on the lads mentality. When you put them over the white line, it's their job to get things done. But when you have those games where you're 2-0 up against Arsenal or Tottenham or 2-1 up against Leeds and the lads don't see it out, what can you do as a manager to perhaps change that mentality or coach them in a different way to see a game out that they haven't been able to do this season so far? Yeah, I think all the situations are different. It's very easy to sort of club them all together and look at them as one. They're very, very different. I think the Arsenal one, the flow of the game did not change at all. They took three of their chances in the last half an hour and they missed them in the first hour. It was basically how the game went. We frustrated them in the same way.  There was deflections. There was bodies on the line. So for me, the flow of the game at Arsenal didn't change. Our game plan didn't change. It's just a long time to hold Arsenal at bay. So never to be good sides. Have a real good chance of scoring late on in the game because they wear you down there. People start to get tired. So yeah, I think the lads understand. They know what it takes to see a game out. Of course, when we get ourselves in front, we try and stay positive. We try and make sure we're still a threat as we were. We were 1-0 up and dango goes through clean on goal 1-v-1 with Ramsdale. I think Aaron Ramsdale makes a real good save from Dom late on at some point in the second half. So it wasn't as if the game changed and we were all 2-0 up. Let's just sit in front of our goal and it was the same. It's difficult for 90 minutes to hold a team of that quality at bay. Up next Liverpool in the Premier League. Everyone will immediately go back to the 9-0 at Anfield. Is that the first thing that comes to your mind or is it the unbeaten six games off the back of it when you were the caretaker manager? Yeah, the 9-0 doesn't come into my mind at all. Obviously, we were in a very different place then. So the lads will just prepare for this. It's another game against a very, very good side. They're in good form at the moment. Obviously, a fantastic attacking threat. It'll be a tough ask, of course, but we approach the game like it's a game that we need to take three points from and that's all. I'm presuming that you would have sat down with your laptop and your notebook watching the Liverpool Manchester United game last week. What was going through your mind in that second half when you were seeing Liverpool demolish Manchester United knowing you've got to play them next? Yeah, a couple of things. I thought it was not a great time for them to hit top form. And then secondly, hopefully, they've used up all their goals as the last swing I was thinking. But yeah, they've changed a little bit recently. They're using a slightly different structure. So yeah, last three or four weeks, they've mixed things up, changed how they played. They probably haven't done that in quite a while while Jürgen's been there. So a slightly new look. But yeah, same sort of threat. Salah on the top line. Nunez obviously a huge threat. Lots of pace up front. So yeah, it'll be a tough ask, but boys are looking forward to it again and an incredible opportunity for us to give ourselves a chance of putting another big result on the board. Thank you. Can I just firstly check on injuries, Gary? Obviously, Jefferson, Harrod Troy were missing last week. Lewis Cook, we saw back. Marcus Tavernier, could you update us on some of those? Yeah, so Tav will still be out. Junior will still be out. Junior Trey Aure. Jeff's touch-and-go. Jeff's not done as much as you would like, but he's further along than the others, so we'll make a call on Jeff tomorrow. Lewis Cook still back. So good to have Lewis still. Yeah, the rest should be fairly similar to where we were last week. How much is availability, I guess, dictating how you go in these games tactically and shape-wise? Yeah, I think we haven't had as many options as we would like, tactically, because of the injury situation, but also the calibre of team we're playing at the moment obviously dictates a little bit tactically as well. Man City Arsenal, Liverpool as a free-game run is not the easiest run you can have. So yeah, there's a few things tactically, but getting players back fit is huge. I've said it a lot this season, but it is. So we need to get them back as quick as we can. We've got Liverpool and Villa and then an international break, which will hopefully help us as well and hopefully come out the back end of that in a much stronger place, numbers-wise. Is that risk, reward, I guess, balance something that you're continually toying with? We see Arsenal last week have 80-plus percent possession, obviously if you camped in your half for a lot of the time. In terms of how you set up, the risk and the reward obviously in games like that are, if you go too open, you're going to get picked off probably. How is that proving difficult for you to decide how to go in these games? Yeah, there's a decision. Of course, Man City, we went very different to that. So we tried to be front foot and aggressive and disrupt their control and structure in a Man City game, especially with it being at home. I thought we could create an energy here and make it uncomfortable for them. Whereas Arsenal, they're an excellent side. I didn't see us causing them a huge problem if we went front foot and energy and I thought the home support and the nature of the game at the Emirates would mean that that would help them. So I felt the best way was to make them go around us, make them throw balls in the box that were less controlled, have numbers around it, scored two goals from the training ground, lead at the Emirates for a long time. So although it ends in huge disappointment, there was a lot of stuff that worked very well. You obviously came here from Liverpool in the under-23s. How much did you learn from your time there? Yeah, a lot, a lot. Yeah, fantastic club. Yeah, really good academy. Produced some fantastic players. Watching the elite level that the first team work at.  Yeah, learnt a lot from, obviously started my coaching career there. So learnt a lot from the guys around it. Yeah, so looking forward to tomorrow. Obviously was there when we lost 9-0 last time, but wasn't the head coach, so yeah, looking forward to it. And did you have much interaction with Jürgen Klopp there? Is there any relationship between the academy and him? Yeah, saw him around a bit. Obviously there was COVID at the time, so it was trickier to move between groups. But yeah, he attended some of the games and saw him around the training ground occasionally. Yeah, that was nice guy, was very helpful. So yeah, looking forward to going up against him tomorrow. Just finally, something you mentioned with Mark about game management and defending leads and things. When does, I guess, losing leads become a concerning habit rather than just one of those things that happen occasionally? No, it's always a concern. You don't want to lose leads. As you guys rightly say, we've lost three this year from two goals in front. So the real disappointing one for me was leads, obviously, because Tottenham and Arsenal are, I remember the Tottenham one, we were fairly comfortable and they lined about £150 million worth of subs up and brought them on. That made a big difference. And like I say, the Arsenal one felt they were strong throughout and it's hard to do what we did for 65 minutes, let alone 97. So the leads one was the disappointing one in that for me.


But nobody cares about when you do that, and there's like muscles there. Nobody cares about that. We don't want to see that.


Today, we are going to discuss the surgical anatomy for low anterior dissection for cancer of the rectum. One of the prerequisites are that the procedure should be reproducible. It means that if I am able to do, then everyone should be able to do. And that is why we are doing this surgery only with monopolar energy source. The first step is to lift up the sigmoid colon. And by doing so, we also lift up the inferior mesentric artery as shown by the red arrow. The first step is to incise the peritoneum, the visceral peritoneum, just adjacent to the inferior mesentric artery at the level of the promontory. While doing so, you can identify the pre-sacral fascia and the mesorectal fascia now shown by the blue arrow. The pre-sacral fascia is the fascia which is underneath the hypogastric plexus of nerves and the mesorectal fascia is the one which is above it. Now with the monopolar energy source, we are dissecting between the nerve and the mesorectal fascia. The purpose is to identify the loose areolar tissue which is now being shown just underneath the promontory. And now you can see the hypogastric plexus of nerves and the cores. The important thing here is being shown is that the right urator is lateral to the hypogastric plexus of nerves. Now the same dissection is continued upwards to identify the inferior mesentric artery and what lies lateral to the inferior mesentric artery is nothing but the left urator. Now you will be able to see with the medial to lateral dissection, the left urator and the left hypogastric nerve as being shown by the monopolar energy source. So you can see the hypogastric plexus of nerve and the urator lateral to it. Now the dissection is continued cranially towards the inferior mesentric artery. You can see the peristalsis of the left side of the urator, clearly the inferior hypogastric plexus of nerve being shown at the level of the promontory. And now you retract with your left hand forceps the inferior mesentric artery and the dissection is continued with the monopolar cotry cranially to identify and dissect the inferior mesentric artery from all the sides. Remember it is very important to stay close to the inferior mesentric artery at this point of time as the hypogastric plexus of nerves converge at this point of time. Now we have opened the left side of the peritoneum from the medial to the lateral dissection. The lymph nodes right up to the level of the root of the inferior mesentric artery are also removed. You can see the left colic branch taking off from the inferior mesentric artery and distal to the take off of the left colic branch the inferior mesentric artery is clipped and ligated and cut. Once you cut the inferior mesentric artery the assistance should hold the inferior mesentric artery pedicle up because just underneath the inferior mesentric artery pedicle would lie the holy plane of healed. This is an avascular plane. As you can see the rectum is lifted up by the assistant and you can see the loose areolar fatty tissue. As you can see in the PIP diagram we are going to focus on this layer of mesorectal fat. The mesorectal fascia completely encases or encloses this fibro fatty tissue which comprises of the rectum and the perirectal lymph nodes. You can see the loose areolar tissue being cut posteriorly. This is the posterior dissection of the mesorectal fascia or of the mesorectum. At 8 o'clock position you can see the vessels being exposed that is where the presacral fascia was breached mistakenly and now the correction of the fascia dissection more proximal more closer towards the rectum is done. The same mesorectal fascia dissection is continued posteriorly right up to the levator and I and posterior laterally also the bipid of the hypogastric plexus of nerve can be seen on both the sides. Now in this view you can see the left hypogastric plexus of nerve being pulled up and in sometime you will also see the right hypogastric plexus of nerve. The mesorectal fascia is continued posterior laterally now as being done all the dissection needs to be done from the posterior side itself right up to the levator and I and in doing so you can identify the buttocks of an infant hanging down as being shown in the video. This is the right posterior lateral dissection being performed. You can see in the PIP mode now once the posterior and the posterior lateral dissection is done the rectum is pulled by the assistant surgeon and the anterior dissection between the urinary bladder or the seminal vesicle and the prostate and the rectum is performed. This fascia is called as the denonvillus fascia. The anterior lateral dissection is very carefully performed because the sexual fibres or the nerves run laterally near the pedicle of the seminal vesicles. You can see the seminal vesicles and the prostate gland coming up. The denonvillus fascia are basically two fascias which are closely in relation. The posterior or the first layer is in relation to the rectum and the anterior layer is more in relation to the prostate and the seminal vesicle. Now if the dissection is performed in this area then we are going to take the second layer of the denonvillus fascia as well and expose the seminal vesicle and the prostate completely. However because this needs to be preserved the dissection is done in the first layer of the denonvillus fascia that is closer to the rectum. Now you can see this is also the envelope of the mesorectum the same envelope which continues from posterior to the posterior lateral and then to the anterior lateral and then finally to the anterior side. We had already done the dissection of the left side from posteriorly that is the left posterior lateral dissection and hence we just needed to cut the peritoneum. The left lateral dissection of the mesorectum is done by the monopolar forceps again. The whole surgery is performed with monopolar just to make you realize that this is a beautiful envelope which is completely avascular and if you remain in the right plane then all the structures can be done by monopolar and you don't really need big instruments like ligature or harmonic. However in 25% of the cases you encounter the middle rectal artery as is being shown in the PIP mode and in the in the surgery the middle rectal artery is ligated and cut with the ligature and this is an ultra low anti resection and hence the dissection is performed right up to the levator and I which is being cut the superficial fibers of the levator and I so that the whole resection can be done right up to the levator and I or the pelvic floor. By doing so we are not trying to cone down the specimen so the coning of the specimen has to be avoided at all cost. As you can see the dissection is performed from all sides the mesocolic fascia is dissected from all the sides that is the posterior the anterior the posterior lateral and the anterior lateral exposing the posterior surface of the prostate and both the seminal vesicles completely. Now this shows the whole of the surgical anatomy or the facial anatomy for dissection of the cancer of the rectum or the pelvic surgeries. Thank you.


As Salaamu Alaikum. Welcome to lecture number 19 of the course on statistics and probability. Students you will recall that in the last lecture I discussed with you various approaches to probability. We discussed the subjective approach and the objective approach. Under the objective approach I discussed with you in detail the classical definition of probability. Also I touched upon the relative frequency definition which is the very important definition from the statistical point of view. Today I will discuss with you this definition in a little more detail and then we will proceed to the more mathematical definition and that is the axiomatic definition of probability. As indicated in the last lecture the relative frequency definition pertains to those situations where we can say that a particular experiment has been repeated a large number of times. As you can see on the screen if a random experiment is repeated a large number of times say n times under identical conditions and if an event a is observed to occur m times then the probability of the event a is defined as the limit of the relative frequency m over n as n tends to infinity. Symbolically we write p of a is equal to the limit of m over n as n tends to infinity. Students in this definition we have nothing to do with the m and n of the classical definition. In classical definition n denoted the total number of possible outcomes of a random experiment and m was the number of the favorable outcomes. But here n denotes the number of times your experiment is being repeated and m denotes the number of times the outcome of your interest occurs. Actually if we want in this definition m or n ki bhajai hum koi aur letters istimal kar sakte the. Now the next thing is and the important point is that this definition assumes that the ratio m over n tends to us tends to become stable as n increases indefinitely. Let me explain this point to you with the help of an example. As you now see on the screen if we consider the coin tossing experiment no one can tell which way a coin will fall. But if it is a symmetrically constructed coin we expect the proportion of heads and tails after a large number of tosses to be nearly equal. An experiment to demonstrate this point was performed by Kerish in Denmark as far back as 1946. He tossed a coin 10,000 times and obtained all together 5,067 heads and 4,933 tails. The behavior of the proportion of heads throughout the experiment is shown in the figure that you now see on the screen. The x axis represents the number of tosses that were carried out whereas the y axis represents the proportion of heads that we obtained for various numbers of tosses. Students, you have noted that this diagram shows that in the beginning there were wide fluctuations. Sometimes the proportion of heads was much higher than 10.5 and sometimes it was much lower but as the number of tosses increased those fluctuations began to diminish and that curve representing the proportion of heads tend to the number 0.5. If you have a look at the diagram once again carefully you see that this trend of the proportion of heads approaching the value 0.5 is very clear as the number of tosses increases and it is quite easily understood that based on this trend that we see we can expect that if the number of tosses were to be increased indefinitely this proportion would tend to the limiting value 0.5. Students, this is the value that this proportion is tending to as n tends to infinity. This limiting value is taken as the probability of heads in this particular example. Let me share with you another very interesting example and this pertains to real life data that was collected in England. This example that I will be sharing with you is related to the sex ratio, the proportion of male births as compared with the female births and the first thing to note in this slide is that ever since the 18th century at least in certain parts of the world it has been noticed that in reliable birth statistics there is always a slight excess of boys as compared with girls. As you now see on the screen, Laplace recorded that among the 2,15,599 births in 30 districts of France during the years 1800 to 1802 there were 1,010,312 boys and 1,05,287 girls. The proportions of boys and girls were thus 0.512 and 0.488 respectively indicating a slight excess of boys over girls. Students, you have noted that the total number of births that were considered were 2,15,599 and that is a large number and the definition that I am discussing with you is that the total number of times the experiment is repeated that number should be large. 2,15,599 this is a fairly large number and if this proportion of male births is slightly higher compared with the female births then we can say that the probability of a male birth is slightly more than 0.5 and you may be a little surprised to hear this because I think generally we think that probability of a male birth and probability of a female birth they are both 50 percent. Of course, if the number of births were smaller if n is a smaller number then we can expect that the proportion of heads will be different from the limiting value as you noticed in the example that I discussed with you earlier in the coin tossing experiment when we had only a small number of tosses the proportion of heads was sometimes much higher and sometimes much lower than 0.5. So, let us consider another very interesting example as you see on the screen we now have a table in front of us showing the proportions of male births that had been worked out for the major regions of England as well as the rural districts of Dorset for the year 1956. If you concentrate on the column of the rural districts of Dorset you notice that the proportion of male births fluctuates between 0.38 and 0.59. In the district of Beaminster it is only 0.38 but in the district of Shaftesbury it is as high as 0.59. On the other hand if we have a look at the various regions of England which are obviously much larger in population than the rural districts of Dorset we find that the proportion of male births fluctuates between 0.512 and 0.517 the fluctuations for the various rural districts of Dorset are much more than the fluctuations for the major regions of England. And the point to be noted is that the figures for the various rural districts of Dorset are based on only 200 births or so each whereas the figures for the major regions of England are based on about 100,000 births each. So students the same point is validated when you have a small number of trials, a small number of repetitions of the experiment, the proportion of the event of your interest will fluctuate quite a lot and it will not be close generally to the limiting value. The value that it will approach only when the number of trials, the number of repetitions of the experiment increases to a large number such as 100,000 in this example. If you have a look at the table once again you notice that the proportion of male births for the entire country is 0.514 and the various proportions for the various regions of England do not deviate much from this overall proportion for the whole country. So I think you will agree that we can assume that the proportion tends to the limiting value 0.514 or something quite close to that because that is the proportion that is valid for the entire country. Though this example says that the probability of male birth according to this definition is higher than 0.5 and it is something close to 0.514. Students why is it that I said a short while ago that this particular definition is the most important definition from the statistical point of view. The reason is that in the real life situation, in the real life world there are numerous situations where we do not have a situation where the various possible outcomes of a random experiment are equally likely.  We cannot say that the thing is so symmetric or that situation is so perfect. That one outcome is as likely as the other. Rather we are faced with this kind of a situation where one outcome is more likely than the other and then there is no way of determining the probability of that event the one that we are interested in except by the definition that I just presented. If we are interested in determining the probabilities of this kind, what is the probability that the next student who will be enrolled in the virtual university BCS program is a first divisioner in his FA or FSC. What is the probability that the next man who walks into this room is a smoker? What is the probability that the next person who walks into this hospital has high blood pressure? Students, you have to be careful that to answer these questions you are not in a position to apply the classical definition of probability.    To answer these questions you will have to resort to the relative frequency definition of probability. You will have to collect data from a large number of persons or objects if it is some objects that we are talking about.  If you are talking about students, then look at the record of the students enrolled in the virtual university BCS program and look at the proportion of first divisioners and then you might agree with me that you can treat that proportion as the probability that the student who is enrolled in the virtual university will be a first divisioner. In the last lecture and in today's lecture, I have discussed with you various approaches to probability. The diagram that you now see on the screen depicts the various approaches and the various definitions. We can say that probability is of two kinds, non-quantifiable and quantifiable. By non-quantifiable I mean the inductive approach, the subjective or the personalistic probability. These are different ways of talking about the same thing and you will recall the example that I presented to you in the last lecture that a panel of three judges is hearing a trial. Two of them decide on the basis of the evidence that is presented that the person who is being accused is guilty but one of them decides that the evidence is not strong enough for him to decide that this person is guilty. The quantifiable probability can be divided into two categories. The classical definition of probability is valid in the situation where the various possible outcomes are equally likely and the relative frequency definition applies in other situations where we cannot say that the various possible outcomes are equally likely. A statistician's main concern is this third definition of probability because in statistics by and large we are dealing with real life phenomena where various possible outcomes are not equally likely. The relative frequency definition says that the subjective probability is the one that we are encountering most of the time in our daily lives. As I said last time, we are taking such probabilistic decisions every now and then but the point of view of statistics cannot come to our work, that definition because students I think we can say that it is next to impossible to quantify the subjective probability. You see you can say that something is highly probable, something is moderately probable or something is very improbable in that context when we are doing it in a subjective manner. A particular painting is very beautiful or it is quite beautiful or it is not beautiful but you will not be able to say that the beauty value of the painting of Mona Lisa is 0.96 or 0.91. Though we are doing subjective probabilistic decisions in our mind all the time in our daily lives but for statistical analyses and for statistical research and all these scientific methods that we want to do to draw conclusions about various phenomena on the basis of real data, we would generally apply the relative frequency definition of probability. I have given you so many different definitions and so many details of probability but students the most mathematical definition is yet to come and this is the axiomatic definition of probability. This was formulated by the Russian mathematician Kolmogorov as far back as 1933 and as you now see on the screen this definition says that let S be a sample space of a random experiment with the sample points E1, E2 so on so on up to En. To each sample point we assign a real number denoted by the symbol P of EI and we call it the probability of EI if it satisfies the following three basic axioms. Axiom 1 for any outcome EI, P of EI is a number lying between 0 and 1. Axiom 2, P of S is equal to 1 where S represents the shore event and axiom 3 if A and B are mutually exclusive events in other words disjoint subsets of S then the probability of A union B is equal to the probability of A plus the probability of B. Students, you have noticed that according to this definition a number a real number lying between 0 and 1 has to be attached to every outcome of the sample space of the random experiment. Yanni for example if it is a biased coin not a fair coin but a biased coin the two possible outcomes are head and tail and according to this definition some number between 0 and 1 will be attached to the outcome head and some number between 0 and 1 will be attached to the outcome tail and according to the second and the third axioms the sum of these two numbers should be equal to 1. Now, why did I say according to the second and the third axioms? Yanni second axiom said the probability of the sample space S in other words the probability of the shore event has to be 1. And if we talk about the probability of head or tail the probability of the shore event then according to the third axiom the probability of head union tail where union represents or should be equal to the probability of head plus the probability of tail. So, Saribhat ka lubeh luwab yeh ke dohi outcomes hai, dono exhaustive hai agar aap unka union lein to puri sample space exhaust ho jatiye. To third axiom ke tahit probability of head or tail has to be probability of head plus probability of tail and according to the second axiom because head and tail are exhaustive. So, this sum has to be equal to 1. Ye itna muskil hai nahi jitna ke aapko pehli defa sunne meh sus ho rahe. I would like to encourage you to study this concept in depth and to reflect on the various points that are the various axioms that I mentioned. And I am sure that if you devote a little bit of time to it you will be able to understand that the three axioms fall in place very nicely and they support each other and we have a proper definition of probability. Or note karne ki baat ye bhi hai ke wo dono definitions jo meh aapko pehli di under the quantifiable definition. Both of them relate to the axiomatic definition. Agar aapka coin fair hai to wo jo number aapka sign keringe head or tail ko they will be half and half. Lekin agar aapka coin biased hai jaya saki meh abhi aapke saad discuss kar rahe thi then you will not be able to assign that number to the outcome head until you first roll that coin a large number of times.  Let us now consider another example. The table that you now see on the screen shows the number of births in England and Wales in 1956 classified by sex and whether it was a live birth or a still birth. As you can see the total number of births in 1956 in England and Wales was 7,16,740 out of which 3,68,490 were male births and 3,48,250 were female births. As far as live birth versus still birth is concerned there were 7,335 live births and 16,405 still births. As you will agree there are four possible events in this double classification. Male live birth, male still birth, female live birth and female still birth. The proportions of births in each of these four categories are 0.5021, 0.0120, 0.4750 and 0.0109. Now a male birth occurs whenever either a male live birth or a male still birth occurs and so the proportion of male birth regardless of whether they are live or still born is equal to the sum of the proportions of these two types of birth. That is probability of the male birth is equal to the probability of male live birth plus the probability of male still birth and this is equal to 0.5021 plus 0.0120 and that is equal to 0.5141. Students, you have noted that in this example but towards the end of the discussion I talked about probabilities. I hope you realize that I did so because the total number of births in this example is large enough for me to apply the relative frequency definition of probability and to regard all those proportions as probabilities. And you will be interested to note that the probability of male birth that we obtained in this example just now is almost the same as what we had in the earlier example. You remember that in the earlier example we said that the proportion of male birth for the entire country was 0.514. If we are interested in finding the probability of still birth students, what we should do is similar to what we did just a short while ago. As you now see on the screen, a still birth occurs whenever either a male still birth or a female still birth occurs and so the proportion of still births regardless of sex is equal to the sum of the proportions of these two events. In other words, the probability of a still birth is equal to the probability of a male still birth plus the probability of a female still birth and that is equal to 0.0120 plus 0.0109 and that is 0.0229. So, in this problem we have noticed that we have found on the basis of real data that the probability of still birth in that part of the world in that particular year was 2.29 percent. Students, let us now proceed to the basic laws of probability and these are the laws that will enable you to solve a number of probability problems in a convenient manner. The first one is the law of complementation and as you now see on the screen, if a bar is the complement of an event A relative to the sample space S, then the probability of a bar is equal to 1 minus the probability of A. Let us apply this law to a simple example. Suppose that we toss a fair coin four times and we are interested in finding the probability that at least one head appears. As you now see on the screen, the sample space for this experiment consists of 2 raised to 4 that is 16 sample points. The reason being that each of the four tosses can result in two outcomes. According to the multiplication theorem that I discussed with you in an earlier lecture, 2 into 2 into 2 into 2 that is 2 raised to 4 that is 16 ways of tossing a coin four times. As you will agree, this rule of multiplication enables us to count the sample points very conveniently. Otherwise, if you start listing them, it is not extremely convenient.  But it is possible that after writing 5 or 6, you get a bit confused. Now, the rule of multiplication convinces you that even if you are not able to list them, the total number of such outcomes is 16. Now, the event of my interest is that at least one head appears.  So, if we apply the rule of complementation that I just discussed with you, you will find that it is much simpler. Jo hamara event hai at least one head, iska jo compliment hai that is no head and it is very easy for us to say ke no head wala to ek hi outcome hai and that is tail, tail, tail, tail. So, we apply the rule of complementation that I just presented. The probability of A bar is equal to 1 minus the probability of A. The probability of getting at least one head is equal to 1 minus the probability of getting no head. No head wala outcome, sif ek hai. So, the favorable outcome is only 1. The total number of outcomes is 16 and so the probability of having no head is 1 over 16, m over n, classical definition. And according to the rule of complementation, the probability of A bar is 1 minus the probability of A. So, the probability of getting at least one head is 1 minus 1 over 16 and that is 15 over 16. The next law that I would like to discuss with you is the addition law or the general addition theorem of probability. As you now see on the screen, according to this theorem, if A and B are any two events defined in a sample space S, then the probability of A union B is equal to the probability of A plus the probability of B minus the probability of A intersection B. This equation can be stated as follows. If two events A and B are not mutually exclusive, then the probability that at least one of them occurs is given by the sum of the separate probabilities of the events A and B minus the probability of the joint event A intersection B. We said probability of A union B is equal to probability of A plus probability of B minus probability of A intersection B. This means probability of A union B. Students, do you not remember that A union B means either A occurs or B occurs or both A and B occur. At least one of the two A and B occurs. Or right hand side, the last term probability of A intersection B. Again, do you not remember that A intersection B represents the joint event that both A and B occur? The way I presented the theorem just now, that is perfectly correct. I will once again repeat it for you according to the general edition theorem of probability. The probability that either A or B or both of them occur. In other words, the probability that at least one of the two occurs is equal to the probability of the first plus the probability of the second minus the probability that both of them occur. This theorem key application or its key examples, we will discuss in the meantime. I would like you to study all that has been discussed today and to practice with various questions that you are able to. I wish you the very best and until next time, Allah Hafiz.


These are potato chips, crisps in England. The chemicals in here are a concentrated source of energy that my body could store for later, or it could burn now to power me to do things that I think I need to do, like mow the lawn. And this is gasoline. The chemicals in here are a concentrated store of energy that I can use to power my lawn mower to help me mow the lawn. So if I were to take my chips, dump them on the driveway, and stomp on them with my big boots, the chemicals, the energy would still be in there, but it just wouldn't be as useful to me, especially if I took my lawn mower and I spread them all over everything. So the stuff is there, the energy is there, but I've made it no longer useful. In exactly the same way, there's now less gasoline in the mower than there used to be. I have burned it. The stuff has gone into water vapor and CO2 in the air, and the energy, a little bit of it, made noise to annoy the neighbors, but eventually that just heated up the surroundings, and a lot of it went right into heat. So if you touched the wrong piece on this mower, you would burn yourself now. So what we see in the real world, normal times, stuff and energy are not lost or made, but they're changed from one type to another. And with energy, we tend to change it from useful types to things that are not as useful and eventually the heat that spreads out and does no good for us. A lot of the history of humanity has been finding concentrated sources of energy and trying to get useful things out of it as we change it into useless heat that spreads around the world. That may give you an idea that we'll come back to later. If we were using sun or wind or hydropower to run an electric mower, I'd be making a lot less noise. I'd be making a lot less heat. I'd be using the energy I bought for what I wanted rather than wasting it.


For more videos on people's struggles, please subscribe to our YouTube channel. Progressive sections in the city of Amherstvoort took to the streets demanding affordable housing on January 30. Activists from various youth and student groups, feminist groups, trade unions and political parties marched for housing rights on the call of Wundrabolt Amherstvoort, a housing rights coalition. The protesting organizations included the Communist Youth Movement, FNB Wackwonde, the Socialist Party, etc. Major cities in the Netherlands have been facing an acute housing crisis marked by higher rents, skyrocketing property prices, evictions and homelessness. Housing rights groups have stated the number of homeless people have doubled in 10 years and the housing shortage has exploded. People with low incomes are affected by social housing being sold, liberalized or demolished on a large scale. Especially for people with disabilities, the elderly, young people and starters, there is a shortage of affordable housing. The housing market has not been made for us but for landlords and by landlords who profit from young people who are getting a start in life. We have been failed and we have been deceived. The Netherlands is not a country of opportunities but a country of opportunism. The Netherlands is not a country of education but a country of empowerment. Various progressive groups have formed coalitions like Wundrabolt in Amersfoort, Wund Protest in Amsterdam, Wund Obstant in Rotterdam and Wund Wurzit in Haag. They are urging local and national authorities to formulate policies to resolve the housing crisis in the country. These coalitions have organized major mobilizations through the last year demanding housing rights but they haven't received any response from the authorities yet. The coalitions have joined forces and released a housing manifesto with various demands to solve the housing crisis. They stated in the manifesto that more than 800,000 households in the country do not have enough money left over for daily expenses after paying rent. The social inequality between renters and homeowners is still increasing enormously due to the disproportionate benefits that homeowners enjoy. The demands include reinstating the Ministry of Housing which deals exclusively with policies aimed at realizing the right to housing, combating homelessness as a top priority, ensuring broadly accessible affordable and secure public housing, expropriation of vacant and unused spaces, and curbing the financialization of housing. The Wundrabolt-Amersfoort coalition stated that rents are unaffordable as a result of political choices and that investors are buying houses and pumping up prices to fill their own pockets. Different housing coalitions have announced protest actions in other cities of the Netherlands in the coming days and weeks.


What is going on guys welcome back to the channel critical overload here's to my scream six in this video here again today Follow up to my Neve Campbell video that I just recently did because now two more I would say more credible reliable individuals known as insiders who Again have a better reputable reputation than me have come out and basically said the same thing I stated a few days ago now viewer and on is the most recent one He stated that yes I've heard Neve Campbell will actually appear in scream six But the bigger news is that the deal apparently calls for a much bigger role in scream seven now as far as like the bigger Role in scream seven I'll touch on that and share my thoughts on what I think is gonna happen based off of that tweet and share my own details about What I was told? After I read this next follow-up tweet So when it relates to one take news one take news I believe is an association with big screen leaks So the fact that they of course went out of the way to make a tweet like this This was then again another thing that shows us we're getting warmer We still don't have the actual confirmation because we of course will remain Unconfirmed until we either see Sydney or Neve's team comes out and says it but at this point Maybe the expectation should be they're not going to announce it because again We're in a no-way home situation and they want you to go out of your way to see it and word of mouth to spread Okay, I can go see it because yes, Sydney was in this movie. So as far as like Neve and that whole deal with scream seven What viewer non tweeted that's exactly what I was told by the individual who I'm telling you I trust because they haven't been wrong with me so far That I was told the same bit of information about a scream seven deal and her having some type of Her having like something in place it already related to scream seven That's essentially what I was told like the gist of what viewer non just put in his tweet as well So as far as like what that larger role will be In this instance what they're saying when they say larger role I believe they are indeed talking about her having more screen time Again, the fact that they went ahead just like what screen five someone pointed out can continue to shoot this movie and Now if she apparently has actually signed on now with her Coming in midway through production for both movies and again with them going on and filming this one actually with her not even confirmed to be in it until Mid until like I would say filming started till after filming started That's a sign to me that she's not gonna have much screen time in this movie She she might have even less because again, they went ahead and started filming screen five without her being around and When she was there, we know we got one of the shortest appearances from Sydney Prescott ultimately in screen five So with screen six, I'm confident. She'll have just Maybe even less screen time. She's more than likely gonna have less screen time than what she had in screen five The gist of what could be going on is you're gonna have your survivor focus movie where you're centered on Kirby Gail Chad Mindy I think I mentioned Sam Gail Chad Mindy Tara Bringing in their mother because we know their mother is expected to be in the movie Christina and all of these new characters That's already a lot you have going on there, you know Throwing Sydney in sure you can throw her in but then of course, you know again We don't want to take away from the survivors. You've already established for the new movie They've already made it clear in some early early a plot descriptions that we've been getting told from press Releases that is going to be centered on those four survivors So it's not even going to be centered on Sydney Prescott, which again, I think is another sign Added on to how she has allegedly signed on after filming it started That her role is going to be even smaller than screen five What she could ultimately end up doing is appearing at the end of the movie Setting up something that leads into screen seven and her larger role is because of the fact that she it's it's something that has to happen At the end of screen six that sets up her having more screen time in the screen seven and I'm not saying they're shifting the focus completely away from Sydney or away from the carpets. I meant to say back to Sydney What they're gonna do maybe is give you a narrative in which the screen Requires everybody on on the screen at the same time with everyone having their own importance to why they are here again in Screen five it really didn't make any sense for Sydney to be there all that much when it's not about her as much as it was a a I guess you could say a scheme to get her back to town and still ultimately wasn't even about her it was about Sam and then Considering what it seems like it's happening with screen six from the early description about the four survivors and you know catching up with them She's she's so far to me doesn't even have a reason to have a lot of screen time in screen six So what they're going to do most likely is Establish a reason for her to have a lot of screen time in screen Seven with the ending of screen six where she will appear She'll do whatever she needs to do to set that up and then there's how her larger role comes into play for screen seven Well, she'll have more screen time You'll get that you'll get everybody like myself who grew up with that original trilogy Getting to see the film shift focus back or at least having Sydney on screen more in the thick of things with a ghost face Killing spree getting to see her either try to solve it Figure what's going on to protect her family all of that stuff very very much so full circle stuff Especially if it's intended to be the last installment She's right there in the thick of things not only with Sydney, but you have the carpenter sisters who I have go of course We'll believe they'll make it out of this movie alive. They seem to be the most Safest in my opinion out of the two I would say general or take is the safest due to due to the impact Jenna has right now in the genre a shout-out to general or take her but Sydney press Scott I'm now even more convinced and have even more faith in what I was told she most likely will appear in This movie if she doesn't then she just doesn't but again with those two people that I showed you at the beginning of this video Coming out and saying it now too. I Have even more faith in what I was told by again, so when I trust and Also considering the fact that those tweets were going on about things about a scream 7 deal that I was told briefly about as well I'm like, okay We're getting warmer. We're getting warmer. We're getting warmer if we get that official announcement. That's fine with me too I just think that what we all should probably prepare ourselves for is it's not gonna be announced if anything Maybe it gets leaked because of someone identifying her on set and Go from there if it happens, but most likely they're gonna try to keep this as under wraps as they can probably and you'll have Sydney press got show up in the movie as a surprise and that of course will lead to word of mouth internet goes crazy Versus, you know blowing your load right now Versus hyping it all up with the issue. Isn't she using that as a marketing tactic? That's honestly probably what they're gonna do And we should pay attention probably to how these cast and crew members react to the question about never going forward because that'll be very Telling as well. Obviously, but let me know just think about this down in the comment section below if you haven't read Of course, make sure you subscribe turn on post occasion You never miss a video in the description. I have links my social media accounts on Facebook Twitter and Instagram You can message me there, of course Let me see any movies deserve you usually let me to cover in the future with all that in mind guys I will see you in the next video


CFA is, we call ourselves a family, we celebrate the fact that we are a diverse organization and we want to make every brigade in all our community open to obviously the youth, to the different genders, to people of different religious persuasion, to some colors, different races. That CFA is a family that is open to take everybody on board. As a means of demonstrating that, myself and the CEO will lead with Russell Wright, a volunteer that's organized the Pride March on the 29th of January, to march down St Kilda to really show the fact that hey, we invite and love the inclusion of in this case the LGBTI community as members of CFA. So it is a chance. So Russell's theme that he's come up with and well done Russell is bring a straight mate. And so a straight mate, I'm happy to march and represent CFA to really show the fact that we're an inclusive organization. If you're gay, if you're straight, if you're male, if you're female, you're all welcome and in fact we love you to be a part of CFA. So I wear the epilote, I'll wear that up until and including during the march to celebrate the fact that CFA welcomes and is an inclusive organization. So my challenge to you is come along, come along and enjoy, join us on the day. Whether you're straight, whether you're gay, whatever your orientation is, please join us and let everybody know that they are welcome to be part of our organization.


All right guys sourdough normally a lot of work very difficult. You have to stay chained to your kitchen for 48 hours straight maintaining the sourdough in order to make a reasonable loaf of bread. That is not the case. I'm going to show you guys how I figured out how to do like a simple minimal time investment sourdough that like any normal working person can do.  Before we get started guys my countertop is a little messy. I try to clean it once a week. If any of you skinny twinks want to come over in a made outfit or girls by all means save me some time and a headache of cleaning myself. So we have our glass bowl zeroed out on the scale. We're going to put in 10 grams of salt. Apparently I have to retrain my workers on how to apply a label without wrinkling it. 10 grams of salt zero out the scale again. 280 grams of glass bottled mineral water. 280 we'll zero out the scale again. Now we need 500 grams of flour. You want an organic bread flour preferably. We have one on my amazon shop and we also have flowers available on frankiesurrangefood.com. But this ratio of water to flour is very important because if it's too hydrated it will not form properly. So here I have my sourdough starter. I left it outside in the higher temperature to make sure it was really active. We just fed the starter yesterday. So you need to feed your sourdough starter one day at least before making the bread so it's nice and active and healthy. You can get your sourdough starter base culture on frankiesurangefoods.com. The top dried out so we got a bit of a hard crust. We don't really want to put that in the bread because it will stay hard and crusty in the bread and you might have chunks in the bread. So just be careful about that. But you can see underneath that crust we have a beautiful bubbly sourdough starter. So we're going to zero out the scale again and put about 70 grams of starter in here. The volume depends on how liquidy your starter is. You know like this is a little liquidy so you want to do like 70-80 grams. If it's even more liquidy you've got to add more. If it's kind of solid you can add less. So I make sourdough bread every three days so this is going to sit on the counter and then two days from now I'm going to feed it again. We have a whole video on sourdough starters if you guys want to check that out. So that's all the ingredients. We're just going to mix everything together. I like using my hands and a silicone scraper on the sides of the bowl. I like wearing gloves so that I don't have to clean dough off my hands for several minutes. And you really want this to be as dry as possible. Yeah even if you just add a little too much water like 280 or 290 grams especially with the higher humidity this time of year in some parts of the U.S. it can really make the dough kind of wet and it won't rise nearly as well when you go to bake it. So we have our dough ball form together nothing crazy maybe 30 seconds of mixing. Now this is going to sit for 15 minutes to 45 minutes depending on your schedule you know you could go have lunch you don't have to be crazy just let this sit for at least about 15 minutes. Has it been 20 minutes perhaps 30 minutes even 40 minutes? We will never know but it is time to stretch our dough out so the water and flour have hydrated together so now we are going to just pull our dough about 10 times and you'll notice at first it was pretty easy to do but as you continue to do it it'll be very difficult to pull the dough apart to the point where you might not even be able to see it's already built up that elasticity after just a few folds. Now I can't even pull it apart. Alright can't really pull the dough apart anymore so we're just going to put it back in the bowl if you have a lid that's great if not you want to put something on this because we don't want any humidity getting into this dough and now we'll put a towel on top so here outside in this sunroom I have my dehydrator this is set to 105 degrees and I'm going to put it on seven hours so whether you come get this in 10 hours seven hours or 12 hours it would only have fermented for about a seven hour period of time the time and temperature here is the most finicky thing if you don't have a dehydrator you're going to need to use a heating pad if it's any less than like 85-90 degrees outside because this needs to be very warm for a pretty long period of time to ferment properly so later tonight we're going to come back and we're going to get this ready for tomorrow. Alright guys it's been about seven hours so the dough has risen a significant amount now if you happen to forget about this outside that's the reason we set it for seven hours worst case scenario you can still cook the bread in the morning it's just not going to look as nice because we're going to shape it right now so here we have a wooden bread basket I think these are called banatons and we're just going to take a metal sieve or any sieve with some flour and dust the inside if you put a light amount it will probably stick alright so we have our dough maybe half risen very sticky we're going to take our silicone scraper and really just make sure to scrape flush against the sides I've done this before with you know putting some oil around the outside of the dough and honestly it's not really that big of a deal you know because these never stick that badly so once we have it mostly scraped off the sides of the bowl we're going to turn it into the banaton you want to be kind of gentle because you know we don't want to remove all the air bubbles so that's all the prep done this is going to sit overnight until we wake up tomorrow and then we will preheat our oven actually here something I don't normally do but what you should do is you should dampen the cloth that you put on top because the top will dry out watch as I pour 50 cents worth of water onto a cotton rag damp cloth on top so it's the next morning it's been about 10 hours and our bread has kind of risen and then deflate a little bit it is a little warm outside but it should be fine you will need a dutch oven for this probably one of the main factors in the bread turning out good because it needs to be really really hot inside of this which you can't get in a regular oven so we got our dutch oven inside and we're going to put this on basically the highest temperature which is going to be 500 this needs to preheat for 45 minutes this is just a piece of parchment paper that I traced so you can put this at the bottom of the dutch oven to make sure it doesn't stick usually it doesn't but just to be safe all right it's been about 45 minutes so when you take off the lid even though it's dry heat there should be some steam coming out all right so since we have that flour in here we can just kind of gently pry this out and put it on top of the parchment paper now if you forgot about it and left it in that glass bowl you know you'd have a harder time taking it out and it wouldn't be as pretty so I'm just going to take a sharp knife and make a slit down the middle this is so that the bread just doesn't split everywhere when it starts rising now we plop this into the middle of our dutch oven the lid goes back on open the oven back in and then this should be like on 450 now so just lower the heat a little bit and we're going to check on this in 25 minutes all right ladies and gentlemen it's been about 25 minutes so we're going to take the lid off the bread looks like it's been rising nicely we're going to pour a little water in the oven you guys can see the steam if I don't burn my face so that steam is going to help the bread rise even more and I did see that the bread's a little brown on top so we're going to stay around 425 degrees so 20 more minutes and the nightmare should be over I have these towels on the floor because this oven is not watertight so when I pour water in here sometimes it leaks out so just keep that in mind all right I got stuck on a phone call almost forgot so hopefully we didn't overcook our bread a little bit maybe a little crispy but looks perfect well that's the first time I've dropped it on the floor but I guess you guys get a better look at it the the bottom is a bit dark so you know we should have kept the temperature a little lower but definitely good really really good perfect crust on top that's not that big of a deal on the bottom but now we just let it rest and have it for lunch you know so for minimal time invested turned out really good but you know if you get the temperature and timing a little better usually get a larger loaf a little more aeration all right our sourdough has been lonely by himself for a few hours on the counter just cooling off you could eat this right out of the oven it's not really that big of a deal so what I'm going to do is I'm going to slice the whole loaf we're going to have some of it for lunch today and we're going to put the rest in the freezer so it stays fresh for the next two days we have it you know it's never really that easy to cut but it's a lot easier the first day yeah so see even with the minimal amount of effort we got a decent crumb you know nothing crazy but decent all right so we got four slices here we're just going to wrap it up in a towel and put in the freezer well for a minimal time invested let's give it a shot not the best loaf of bread I've made but very good yeah so for the amount of time we put into this bread it turned out incredibly well you know less than five minutes of total prep and the only annoying part is you know that next morning you kind of have to be home for two hours to preheat the oven and cook it but very very little work compared to all the other sourdough recipes so hopefully you guys enjoy this you can get a sourdough starter on frankiesfreerangefoods.com and as I said earlier we have the video explaining how to maintain that starter so you can do this whenever you want as always if you guys can drop a like on the video leave a comment down below my brain's fried today make sure to subscribe and check that notification bell and if you guys want to support me further frankdestifon.com to see all of my interesting businesses I'll see you guys soon


A volcano continued erupting early on Monday in southwestern Iceland, sending semi-molten rocks spewing toward a nearby settlement for the second time in less than a month. The eruption, which first started on Sunday morning, came after a swarm of earthquakes near the town of Grindivik, the Icelandic meteorological office said, the community was evacuated, Iceland's RUV television reported. Grindivik is a town of 3,800 people about 50 kilometers southwest of Reykjavik, Iceland's capital, Iceland. Which sits above a volcanic hotspot in the North Atlantic, averages one eruption every four to five years.


So are we truly able to multitask? Is it true or is it a complete BS myth? In this video, we're going to break down multitasking via science and also talk about what you should be doing to increase your productivity. Let's get into it. All right guys, welcome to the MD journey channel dedicated completely to help students succeed on their journey with less stress. My name is Lux. I'm internal medicine physician and resident and training have been making videos just like this and we're releasing two videos to help students and people just like you. So if you are new to the channel, consider subscribing and hitting that notification bell as well as hitting the like button to help this video as well as this YouTube channel grow. Today I wanted to make a video all about multitasking and if there's a validity to it, it's true how you can do it and if it's completely false what you should be doing. An idea of multitasking has definitely come at risk in the over the last few years and people question whether it's actually valid and possible for a human to focus on more than one task at a time. Let's see what the science says. So initially it's been thought that our brains have enough capability to do multiple different complex tasks at once and we can think of many different examples of this. You know you can walk down the street while texting on the phone, although you shouldn't be. You can read a book while listening to the music. You can talk to somebody while you're working out. You know one of my favorite examples is think about when you're driving and then you become engrossed in your thoughts or a conversation or the music and suddenly you realize that you've made it to your destination.  So to understand the idea of multitasking or just doing a task in general, think of our brains kind of like a factory where you're going from idea all the way to creation. Each step along that journey from idea all the way to completion requires a different process that happens at various portions of our brain. So there's kind of a little bit of a checkpoint. If you think of each of these processes kind of a checkpoint, there's a little bit of a time delay for the part of our brains to process what the information is and then move it forward to different parts of our brain or our body. So for example, if there's the idea that you want to move your right arm, there's a thought that goes into your brain that sends signals to various portions and eventually gets into the muscle fibers in your fingers and your arms to cause the movement. At the same way, when we look at each task from the moment that we tell ourselves we want to do it to the time that we complete it or actually start executing it, there is a time delay at each process. And it's believed and it's been shown in a study in Science by Coach Lin and his team, where basically that a human may be able to process one idea into completion for a task. Maybe even two, they found some individuals in their study who were able to do two different tasks and use two different parts of their brain without interfering with each other. As soon as they added a third task into the mix, their brains simply weren't able to keep up and their performance decreased dramatically. So is multitasking possible? Probably in some sense, but probably also not in the sense that many of us think in the terms of productivity. So for example, many of us think that we may be great at multitasking when we have our emails, multiple tabs as well as our score opened up on our home computers. And while you may be able to go from your email response to going back to studying to rechecking your email and social media, you're probably not multitasking, but instead doing what we call multi switching. And a good way to understand is is now think of your brain as kind of like a roller coaster where each diversion of your attention is kind of a movement of a left or a rapid movement to the right for your brain and your focus. Similar to the roller coaster, it's not smooth. It's not natural. Your brain actually requires a good amount of energy to divert the focus you currently have and somehow create it for your new task. And in fact, they found that individuals who are multitasking can actually spend 40% more time focusing. So while you may be able to get some things done, you actually may be dropping in your performance because you have to regain your attention and your focus to the initial task that you're working on.  I thought I was productive because I could do multiple things at once, but I realized that multitaskers actually have a tendency of being less productive, spending more time and also have less performance and retention on things such as exam. And this goes back to the idea that again, we have to spend so much extra time focusing when we divert our attention in our mental roller coaster from left to right or task to task. There's actually a phenomenon called the Duning Kruger effect, which essentially says that sometimes we give ourselves a little bit more credit than we ought to get. So sometimes you may consider yourself to be more productive because you're multitasking when in reality, your performance actually may be lower than if you chose to do each of those tasks individually. So when we break the science down, it looks like multitasking overall and the desire that we have to do multiple things at once is probably a myth. But now the next question is what should you do and how should you fix it to increase your productivity? Now, if you're somebody who likes to move from task to task, that may be an indication that perhaps you're easily distracted or disinterested in the current tasks that you're working on. So a good recommendation would be to actually shorten your work hours for each specific task. So instead of maybe 15 minutes at a time, you may want to say, I'm going to work on my next study project or a project for work for 15 minutes. I'm going to give myself a five minute break and somehow divert my attention. Now this process is going to require you to take your big tasks like a school project and break it down into something that may be manageable in 15 minutes.  So make sure that the tasks that you pick for these shorter time blocks are actually more appropriate. Now tip number two to improve your productivity and improve how much you get done is to try to reach peak performance through dedicated practice. Now one of my favorite books that I read on this process is inspired by one of my favorite authors, Cal Newport. It's actually a book called Peak, which essentially looks at different individuals. This includes high performers like famous musicians or athletes. Basically asked the question, how did it get to being so good in their time where it looks like almost like they have a natural gift. And they found that these individuals became so good by focusing on one small skill at a time over a long dedicated period of focus. So for example, take the basketball player Ray Allen, which is considered to be one of the best three point shooting and shooters in history of basketball. And most people have this belief that a shooting form is actually a God given talent or something that he was born with from day one. But in fact, Ray Allen himself will tell you that he was actually a terrible shooter in his early days. And the reason that he's one of the best is because he spent focused periods of time throughout his career and his various forms of training to get better at that one skill. And the reason I bring this up is not to help you become a Hall of Fame basketball player, but instead to teach you idea that if you instead diver, I'm doing multiple tasks at once to instead doing one skill that you really want to master such as perfecting the knowledge for an upcoming exam or perfecting a skill for your current occupation or job, you can truly see the most benefit and performance in a short amount of time. And tip number three, if you're having trouble multitasking that may indicate that you may have to do a better job and find an effective system to improve your time management as well as planning. This way you can instead take each of those tasks that you would divert your attention to and have a specific block throughout your day and week dedicated specifically to it. Now personally teach a concept called time blocking and one of my productivity courses called time mastery, which is made for students on the MD journey.com, which you guys can check out if you want more help with your time management as well as improving your efficiency. But understanding that whatever system that you pick, taking the steps to making sure that every bit of time in your day and your weeks are scheduled will allow you to avoid desire to jump from one task to another. With that being said guys, hopefully you guys enjoyed this video saying about the truths and myths of multitasking and saying ways that you can be more efficient and working kind of with your brain's capability of handling multiple tasks. And before you leave, if you did enjoy this video, make sure again you hit that like button if you haven't done so already. We're putting out two videos just like this one on a weekly basis, so go ahead and also consider hitting that subscribe button as well as notification bell. And like I mentioned, if you are interested in learning more about our courses or our blog posts, go ahead and check out the links down below. But I'll also go ahead and link one of my favorite playlists to increase your productivity as a student. Hopefully that helps you increase your efficiency and get more done without the need for multitasking. With that being said guys, thank you guys so much for making it to the end and watching this video. Hopefully I've been a little help to you on your journey. Thanks for being a part of mine. See you guys in the next one. Peace.


Hi, I'm Zor. Welcome to Indizor Education. I would like to talk about one particular example where implicit differentiation is very, very handy. A couple of examples were in the previous lecture where I explained what exactly is implicit differentiation. And this is just one of the examples of some classical and simple function where it's very useful, actually, to have this particular technique. Now, this lecture is part of the course of Advanced Mathematics presented on Unizor.com. That's for teenagers and high school students. I suggest you to watch this lecture from the website because it has explanation, like a textbook, basically. It's notes for each lecture, very detailed notes. And in addition, there is certain educational functionality in the website. If you are signed on, then you can, for instance, take exams, which is very important as a basically development of your logical thinking, creativity, etc. The site is free, so you can take as many times exams as you want. Alright, so let's concentrate on this particular problem. So, my problem right now is to take a derivative of the function arc sin of x. Sometimes notation is sin to the minus one of x. I prefer not to use this notation because sometimes it means arc sin, sometimes it might actually mean really sin to the power of minus one, which is reciprocal. Alright, so let's forget about this. I will use arc sin. Now, before going into the detailed calculation, let's just talk a little bit about the function itself. Now, the function is the reverse of the sin, obviously. Now, the sin has a graph, as we know, something like this. And obviously, there is no real function which is defined for any reverse function. First of all, what is the domain of this inverse function? Well, the sin can take values from minus one to one. So, basically, this function has domain from minus one to one. Now, what are the values of this function? You see, the function sin is periodical, which means the same value of the function corresponds to more than one values of arguments. Actually, infinite number of arguments have exactly the same value of sin. So, if we want to talk about reversing the sin, which means from the value we should really get the argument, we really should restrict what kind of argument we are willing to allow as values of this function. So, the function basically gives you one concrete value for the argument. So, in this case, what is the angle sin of which is equal to x? Well, as we have many different angles. Now, traditionally, what people do, they just cut from minus pi over two to pi over two. Only this interval of angles are allowed. And as we see, as angle is within this particular interval, the value of the sin goes all the way from minus one to one. And this monotonically increasing function, which means it has the reverse function, which is arc sin. So, the values of this function are actually in the interval of minus pi over two to pi over two. And this is the graph of the function arc sin from minus one to one is domain. And from minus pi over two to pi over two, it's the values of the function. And as we know, the graph of this inverse function is symmetrical to the graph of the original function sin relative to the bisector of the main angle. So, at bisector of this at 45 degrees and if you will invert, you will get the sin. Okay, now we are ready to talk about derivative. Now, traditionally, how we derive the derivative for function sin, for instance, or for any other simple function like y is equal to x square or something. Well, we basically follow the definition of the derivative. Now, the definition of the derivative if you have to take increment of the function at point x divided by increment of the argument and have the limit as increment of the argument is infinitesimal value. Well, and we actually calculated exactly what is the derivative of the sin. Now, you can actually, if you want, you can review this, it's one of the previous lectures. And one of the very important points was the limit which we actually proved before that this, as x goes to zero, this goes to one. So, this is the base of the formula which we have derived for derivative of sin. Now, here, unfortunately, we cannot really do something in this particular way because if I will do increment arc sin of x plus delta x, well, it's kind of difficult to deal with. Just think about, how would you deal with this? I have no idea how to deal with arc sin of x plus delta x. When it was sin of x plus delta x, I know what is the sin of the sum of two angles. There is some kind of a formula I was using, et cetera. There is no formula for this. So, the whole method of direct calculation of the derivative might not really work here. At least, I don't know how to do it. However, there is a method of implicit differentiation which does help in this particular case because what does it mean that we have this function arc sin x? It means that the sin of arc sin of x is equal to x. That's what actually means that the value of arc sin, what is it? It's an angle sin of which is equal to argument x. That's the definition of the arc sin. So, this is just the definition of the arc sin. I mean, it falls from the definition of the arc sin. Why is this better? Well, here is why. Now, I will use this principle of implicit differentiation because this is the function and this is the function of x. For any x which belongs to this interval from minus 1 to 1, this thing is true. So, we have two functions which means I can take their derivatives and derivatives must be the same. If functions are the same, derivatives must be the same. Now, derivative of the left spot, well, this is a compound function and you remember the chain rule. If you have a compound function, first you have to take the derivative of the outer function. Derivative of the sin is a cosine of the same argument and multiply by derivative of the inner function. So, the derivative of arc sin of x, I will use prime as a symbol of derivative. And what is it equal to? What's derivative of x? Well, derivative of x is 1. Now, look at this. We basically have unknown derivative which we can determine from here. So, arc sin of x derivative equals to 1 over cosine of arc sin of x. Well, number one, I can just leave it as it is. Number two, I know that this is supposed to be somehow simplified. Let's just think about this again. Arc sin is an angle sin of which is equal to x. So, I know that the sin of this is equal to x. Now, what is the cosine? Well, obviously we know that the sin of square of phi plus cosine square of phi is equal to 1. So, cosine of the angle is equal to square root of 1 minus sin square of phi. Actually, technically I have to put plus minus here. Now, let's think about it. What is this f? It's this. Phi, I should say phi, Greek letters. So, I know that I am from minus pi over 2 to pi over 2, right? So, phi is from minus pi over 2 to pi over 2. Remember the graph of this cosine? This is minus pi over 2 and this is pi over 2. So, it's positive here, which means I can just take only the plus sign. And now I can have it equal to... So, instead of cosine of this, I will just put this. Square root of 1 minus sin square of arc sin x. But we know that sin of arc sin is equal to x, right? So, it's 1 minus... So, the answer is 1 minus x squared. So, that's my final formula. So, this is a derivative of the function arc sin. Well, it might be a little unusual, right? You remember from sin, you have cosine. From cosine, you have minus sin as a derivative. Well, you might expect that the derivative from arc sin must be something like arc cosine. I don't know. But anyway, this is not arc sin. It's completely different formula and that's actually what it is. There is nothing to argue about this. If you remember, the derivative of the function logarithm x is 1 over x, which is also kind of unexpected. But again, it is where it is. So, that's the answer. I do suggest you to go to the Unisor.com website and just read whatever the proof of the same thing actually is. It will probably help you to better understand this principle of implicit differentiation. So, sometimes the functions like this, and you can obviously do it yourself for arc cosine or anything like that. So, sometimes this implicit differentiation is very useful and it looks like otherwise you cannot really derive this formula. At least not easily. At least I don't know how. That's it for today. Thank you very much and good luck.