Probability Distribution Function || Examples & Real-Life Applications

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[00:00:10]All right. I hope a beautiful day. How you guys doing?

[00:00:34]Welcome everyone.

[00:00:38]We have uh a couple of questions to react with today and uh as usual I would love us to participate. I would love to hear your responses in the comment section and uh you guys need to understand that today we're going to be delving into real life applications on the various probability distribution function that we have been dealing with in the past few days. So uh you'll be seeing questions that I'll be throwing at you and uh it will be your prerogative to decide which one follows what distribution. So I will not be telling you anything on that. You would have to use everything I have taught you the features of each of the distribution. You use it to identify which one we're dealing with and follow through on the parameters you're supposed to use and get your result. So, it's going to be a very interesting class. We have quite a number of questions to solve. Actually, we can't solve all the questions that I have available. So, we just try to solve as many as possible. All right. So, let's get right into it.

[00:01:50]Here's the first question. Of course, I will be talking and uh you can move with me uh listening to me, but I would also try and write it down so that in case you want to look at it on the board. All right, the first question.

[00:02:13]Again, don't expect me to tell you this follows a geometry, a poison or a binomial or a benoli.

[00:02:23]U probably I just give you do you do you want me to give you uh a quick revision or are you guys good with all the four distributions?

[00:02:34]Okay, I will try not to give you the revision first because it the questions will help uh remind you of all all the various things we said in previous classes.

[00:02:48]So the first question according to statistics.

[00:02:56]So according to statistics statistics Canada live table.

[00:03:12]All right.

[00:03:15]Think I would The probability that a randomly selected 90-year-old 90-year-old Canadian male Canadian male survives for at least another year is approximately 0.82.

[00:04:12]All right.

[00:04:15]Now if 20 if 200 year old Canadian males are randomly selected. Hey, so the question now is what is the probability exactly 18 survives for at least another year?

[00:05:12]All right. So this is the question we have right in front of us.

[00:05:22]The question said according to statistics Canadian life Canada life table the probability a randomly selected 90-year-old Canadian male survives for at least another year is approximately 0.82. 82. Now if 29 male are randomly selected, what is the probability exactly 18 survives for at least another year? So the first question is what distribution does this question follows?

[00:05:58]What distribution does this follow?

[00:06:08]That's the question. The first question is what distribution does this follow?

[00:06:13]All right. Somebody said binomial. Any contrary opinion to that?

[00:06:18]And the question is why is it binomial?

[00:06:20]What are the features of binomial that makes you believe that this particular scenario is a binomial distribution?

[00:06:29]Somebody said poison. Another person said binomial.

[00:06:33]So which are we going to pick? Somebody said poison.

[00:06:39]Right. Do we have any other more than one trial?

[00:06:44]Somebody said more than one trial. Of course. More than one trial. Any other opinion? Okay. Now guys, we had two different answers coming from the chat.

[00:06:54]But if you look through on this now somebody said poison because of a fixed interval which is 90 year somebody said more than one trial each trial has two possible outcome. Now let me give a response to both answers. This question said the probability that a randomly selected 90 year old Canadian male survives for at least another year approximately 0.82. H2.

[00:07:24]Now somebody the person that talked about this being the poison is taking this is assuming this is the average rate.

[00:07:35]But if you recall what I told you about poison distribution I told you that poison distribution has a known average rate which we call lambda. And I told you that the way to sense those things in question is that you'll be seeing things like number of accidents per day, number of flowers per square meter per something per something. You will be seeing those kind of scenarios that depicts average rate. All right. Looking at this question right here, I do not see any scenario that has to do with a known average.

[00:08:17]In this class, I'll be explaining to you a scenario where the question will look so much like a binomial distribution, but you would have to use a poison distriution. So, you give me just some few minutes. I'll be getting into that very soon. But looking at what we have right here, this is a binomial distribution.

[00:08:36]This 90 year old you have right here is dormant. It's not anything. You're not going to be using this at all in the question. It's more like saying I post a coin 20 times. Now that coin there is the subject. The same way in this case 90 year old is a subject. It's this 90 is not a data you are going to be using in this question. All right? This 90 year old is a subject. You see we said 90 year old Canadian male. It's a subject.

[00:09:07]It's not a data you're going to be extracting. So 90 right there is not something you want to be picking. So these are the questions that they put in there for you to get confused. Use of the word 90 there is not for you to be thinking you're going to use 90 for anything there. 90 is just a default subject there. more like imagine they could have said the probability that a randomly selected lawyer survives for at least another year but here the subject is not lawyer the subject is not a doctor the subject was used as a 90-year-old Canadian male all right now why is it a binomial look at it the priority randomly selected at least another year is approximately 0.82 A2 this obviously probability of success is what we have here because they saying survival of a 90 year old probability 0.82 that means the probability of success which we call P is equals to 0.82 and obviously Q which is 1 minus P will be 1 - 0.82 82 I believe we all know that Q year represents the probability of failure that means probability that the 90 year old people Canadian male will not survive for another year will be 1 - 0.82 A2. Now this is one of the main reason why this is going to be seen as a binomial.

[00:10:33]You see right here you can see if 20 I told you this in in one of the classes we had last week. If 20 you can see in this place now we 20. This depicts the number of trials.

[00:10:49]I told you guys already that trials can come in this form.

[00:10:54]So it's more where let's say we had uh let's say we had 10,000 Canadian male in general we are picking 20 of them we get we got to the place where Canadian males were we we picked the first one that's one trial the second one that's the second trial third one third trial until we picked 20 so the question is now saying 20 of them we have picked what's the probability that exactly 18 of them exactly 18 survives for at least another year. All right, this is where you would now be using all your understanding of binomial probability that exactly 18.

[00:11:37]Now in this case now you can see that your random variable X will be representing the number random variable X here represents the number of 90 year old Canadian male that survives for another year that survives or another year.

[00:12:10]All right, I'm sure you understand that.

[00:12:12]So your random variable X here is representing the 90 year old Canadian male that survives for another year. I told you guys already I've said so much about the definition of random variable.

[00:12:21]So in this case now what they asking you to find right here when they say what's the probability that exactly 18 survives. This is the notation that you will use to answer that question. the probability that your honorable X was equals to 18. That means we said X represent the number of 90 year old Canadian male that survives that year.

[00:12:44]So probability that exactly 18 that's probility that that random variable X was equals to 18. We said X is the number of 90 year old. I told you guys already that the random variables will will always be reflected based on the question you're dealing with. A random variable can be the number of heads you see when you toss all five times. A random variable can be the number of male student you are able to select in a jub's class. It can be anything depending on the question. In this case, your x is representing the number of 90 year old Canadian male that survives. So probability that that x is equals to 18 is what you are told to find. All right. Now, this brings us to what we call the PMF of a binomial distribution. All right. The binomial distribution. I believe everyone should be able to recollect that is we stated this in our last class on binomial probability of random variable x being equals to a particular value is equals to n which is the number of trial combination x all right p raised to power x 1 - p which is q rais^ n - X and we stated that the values for X between zero up until N all this in that class. All right. So you would do really great to ensure you are very solid in the basics. As I always say, the basics is key to everything you want to do. Now guys, what you're going to do for me now is to use this formula to find this question.

[00:14:44]So in this case, obviously what you're going to do now is to represent your x with 18.

[00:14:56]So we're going to be saying your number of trials here what? 20. The number of trial is 20. So your n is equals to 20.

[00:15:06]All right.

[00:15:10]So we see P of X = to 18 will be equals to what? 20 combination 18.

[00:15:19]What's your P? Your P is 0.82 raised to the power of X which is 18.

[00:15:27]And you have 1 - 0.82 raised to the power of what?

[00:15:34]We have two. That's 20 - 18 2. So, can you go on to solve this and tell me what you got?

[00:15:53]So you have uh good you have to learn here is the ability to know how to bring out your data point.

[00:16:09]All right. So I think when you all decided to solve this answer you got I think I can see some answer in the comment section.

[00:16:20]people are writing 073.

[00:16:26]All right. So if you an example, this is exactly what you would have to do to get your full mark. The major thing is for you to be able to bring out the data.

[00:16:41]And you can only bring out the data if you really truly understand all these different uh aspects of the distribution. What P represents, how to bring out Q, number of trials, and also the random variable X. All right, let's see another question. Let me see how you fair with this.

[00:17:06]Are you guys good with this? Did you really understand it? How do you feel?

[00:17:34]Okay. All right.

[00:17:37]That's good. Good.

[00:17:43]So uh you have to be very uh sand in you have to be very sound in bringing out your values. Now the next question which I believe you should be able to solve this easily.

[00:18:07]Next question is for 30 million naira.

[00:18:13]I'm sure everybody's eyes will be opened right now. Those people that were sleeping, they are very much awake. The sound of 30 million.

[00:18:23]Okay.

[00:18:25]So, uh by the way, take that on favor. I don't you to hold me for 30 million.

[00:18:34]Okay. So, uh, a multiplechoice test contains 20 questions.

[00:18:54]All right, we answer choices A B C and D.

[00:19:09]Now only one answer obviously will be correct. Only one, sorry, only one answer choice to each question.

[00:19:28]Which question is correct?

[00:19:32]As usual and as expected, represents a correct answer.

[00:19:43]So the question is for you to find the probability that a student will answer exactly six question.

[00:20:18]Now if he makes look at the part that is very important and there's a reason why this part is included there. I will explain it when you guys are done making your attempt.

[00:20:42]Now look at this question.

[00:20:44]A multiple choice test contains 20 questions with answer choices A, B, C, D. Only one answer choice to each question represent a correct answer.

[00:20:58]Now to find the probability that a student will answer exactly six question correct if he makes random guesses on all 20 questions.

[00:21:13]So what's your take on this?

[00:21:21]Also the first thing you need to you need to first be able to decode what distribution are we dealing with here.

[00:21:39]So you need to be able to tell the distribution you are dealing with. So guys that's the first question. What distribution is this?

[00:21:56]So is this a geometric distribution?

[00:22:13]Is this a poison distribution or is this a binomial or is this a benoli?

[00:22:29]You need to be very sound in being able to make this correct pick. If you really want to be able to answer the questions, so what's your answer to that? Somebody said benoli. Somebody said binomial.

[00:22:48]Said benoli.

[00:22:55]Okay. Somebody said I don't think it's binomial.

[00:22:59]Somebody said benoli. Somebody said boli.

[00:23:03]Somebody said binomial.

[00:23:06]You understand the main reason why we are solving these questions.

[00:23:13]Somebody said binomial. Okay. Nobody is saying poison. Nobody is saying geometric. All right. Now let's talk quickly about why this cannot be a Benoli distribution.

[00:23:26]All right. I told you guys that Benoli distribution involves only one trial.

[00:23:34]So once you see any question that involves more than one trial just understand that that cannot be a benoli distribution. It doesn't matter whatever you see there even if they told they wrote a word Benoli because I want you to understand that they used to sometimes state write binomial as n benoli because as I told you already binomial and benoli are very similar. The number of trials is what differentiates them in almost all case in all cases. Now if they say n benoli obviously this is not b this is binomial this is not benoli to be very uh open to that now why is this binomial look up I told you guys that I'm going to explain why this last thing they wrote here if he makes random guesses on all 20 questions.

[00:24:38]Now first of all you can see that this is 20 questions. So obviously he's having 20 trial is having 20 trials.

[00:24:50]So it cannot be this can never be a Benoli. If it's just one question that's when it's going to be Benoli. All right.

[00:24:57]Now the probability that student will answer exactly six question correct correctly if he makes random guesses. If this student is not making random guesses on all 20 question, this is not going to be a random variable. I told you guys already that the use of the word random when we say random variable is number one the fact that random comes from the word probability.

[00:25:26]There must be a an unpredictability in that question in everything you're doing when you're dealing with random variables. Once you are assertaining of a particular result, you're no longer dealing with random variable. It's no longer by chance. So if this student knew the answer to any of this question, question one, let's say he knew what question one was, he knew that the correct answer was a, he knew that if he knew that this is no longer a random variable question.

[00:25:58]That's why this was very essential to be included in the question. They said if he makes random guesses meaning that the pick on every of the question was random which is what makes it a probability question a probability distribution function. All right note that one somewhere. Now let me tell you the other reason why this is a binomial. So you can see already that of course this is 20 questions.

[00:26:23]Now 20 questions with answer choices A B C D. I told you guys that the feature of binomial is that binomial has n trials.

[00:26:34]Number one, you can see this is 20 trials. So this is ticked. Number two, it has two possible outcomes.

[00:26:46]Two outcomes. Exactly two outcomes.

[00:26:49]Success or failure.

[00:26:53]If you look at this, there is only two possible outcomes here.

[00:26:59]If the student picks option A is ether is successful or not. So among this ABC look at only one answer only one answer choice represents a correct answer. Very important very very important. All right. So let's say question number one now option A is the correct one. Let's assume question number one. A is the correct one. Number one, we had A, B, C, D.

[00:27:33]The probability of success for question number one. And for every other question will be what? Can anybody tell me in the comment section what's going to be probability of success?

[00:27:45]P. That's small letter P. And we will talk about small letter Q. Probability of failure. What's going to be the probability of success?

[00:27:56]What's going to be the probability of success?

[00:28:00]Can I get great answers there?

[00:28:04]What's going to be the probability of success? The probability of success guys will be one out of four because we said one answer choice to each question represent a correct answer. So, probability of success here is 1 / 4 and probability of failure is 3 over 4. We only have one option that is correct. So success if he picks A may be successful.

[00:28:29]If he picks if A was the answer if he picks B. So it's the two possible outcomes here is just as I explained to you when we're dealing with dice. I said if five is what you want to see this success is when you see five. The failure is not five when you don't see five. So every other numbers you see not five is seen as failure. So if a is the correct answer here and he picks any other one all these are seen as failure.

[00:28:57]So all these three if this one if a was correct one all these are failure why this a is success. So probability of success here is one over four. All right 1 / 4. So that's the way you tackle questions like this. You need to be able to pull out all the data. All right.

[00:29:19]First of all, you must be able to identify what kind of distribution you're dealing with. And since we have been able to detect that this obviously follows a binomial because we saw end trial number one, we saw two possible outcomes. He said only one answer choice to each question represent the correct answer. So two possible outcome is either he got it right or he failed it.

[00:29:39]If if he chose a well while a was correct that means he got it right. If he chose any other one apart from that a that was failure.

[00:29:47]Now the next thing we need to talk about is what does our random variable here represent?

[00:29:54]Can anybody tell me what our random variable represent? Because I want you guys to be so s in this topic that nothing can can give you any issue in any exam you're going to sit for. You need to know this very well. Our random variable X here will represent what?

[00:30:16]X will represent the number of correct questions that this student has. We find that a student will answer exactly six questions. So X here represents the number of questions correct or the number of questions that he got correctly. It represents number of correct of of questions he got correctly anyhow you want to frame the English you understand that doesn't have to be word for word all right you can have it uh with a different the same saying the same thing but of course different odds. Now, now that you know this, obviously you know that we already know this mirrors a binomial distribution and you know the PMF for a binomial distribution and as of as always you understand I would love to get you guys so familiar with this by rewriting the PMF as often as possible. So probability of the random variable being equals to X which in this case now look at the question here we said find probability that a student will answer exactly six question that means probability that the random variable X is equals to six of course random variable X don't forget represent number of question you got correctly and that was what was being said here the way you're going to be able to pick your random variable X knowing what it represent is based on the flow of the question so the question find that a student will answer exactly six questions so Now you know already that correctly answered question is what our random variable is mirroring. It's very simple. You just need to understand the basics. All right. So our PMF is obviously N combination X ra^ X and Q^ N minus X. X ranging from zero up until N.

[00:32:09]And I explained this to you before. Your values for X your random variable cannot you cannot have your random your number of questions you got correctly now to be 21. It's not possible.

[00:32:20]So your value for your random variable will be between zero. It may it may get all the questions wrong. You may have zero zero correct answer. There are some people uh some students that their random variables will always be producing zero. They didn't get nothing.

[00:32:37]But some used to get everything. But there is nobody there's no matter how smart you are that you have 20 question, you're going to get 21 or 22 or 25 out of 20. It just doesn't work. So that's why your random variables value cannot exceed n is between zero and n.

[00:32:54]So right now we have uh this and I believe you should know how to solve this because the n trial we told you already is 20. So we have 20 uh combination six probability of success we called it we remember we got that to be 1 / 4 which is 0.25 25 raised to power 6 then 0.75 which is Q 1 minus of course 3 4 is 0.75^ 20 - 6. So can you guys solve this and tell me what your answer is.

[00:33:27]So this is we're trying to apply binomial expansion and every other sorry binomial probability binomial distribution and every other distribution as much as we can this night to real life situation to modeling real life experiences. So guys what's your answer when you are when you attempted this what did you get?

[00:33:51]If somebody said the answer of uh the answer was 0. What was the answer I saw right now? Is it 16? Is it 0.16?

[00:34:07]Somebody said one. Why you having different answers? Somebody said 1686.

[00:34:12]Somebody said 1669.

[00:34:14]Someone said 0.17. I told you guys already. Those of you writing 0.17 I think you have done AC approximation as much as possible always try and so 0686 is what I would like to write and then later on you can go on to approximate this to 0.17 but make sure you write this first when you are in an exam all so guys how do you feel are you better do you enjoy it are you enjoying Okay. So let's see how this goes.

[00:35:18]Somebody's asking for 30 million after I've explained for almost minutes.

[00:35:27]You should give me 30 million. I think I'm the one that sells 30 million here.

[00:35:32]Don't you guys think?

[00:35:37]So this is how you are going to this is how you're going to deal with questions on all this when you you're in an exam hall. Make sure you're confident. There's nothing in any of them. You can do it. to make sure you make sure you just stay very precise, understand what you're dealing with and uh and you move through. Look at this question. Now we have 25% of all students.

[00:36:14]Again, the first thing is for you to detect what kind of distribution you're dealing with. Nobody will be telling you enrolled in high school XY Z.

[00:36:37]So an ice school is called XY Z.

[00:36:58]Okay. So, uh we're going to check up if this is if this is a poison, a binomial, a bin or a geometric distribution.

[00:37:27]So you see this question now. We're told that 25% of all students enrolled in high school XY Z are taking algebra.

[00:37:49]All right.

[00:37:52]Now the question is for you to you know they said 30 students are chosen at random. We have to find the probability that exactly seven students out of the 30 chosen at random obviously are taking algebra. So guys you can start solving that one. That's the first question.

[00:38:36]So, have anybody got an answer yet?

[00:38:51]Any answer yet?

[00:39:16]All right. So if you look at what we have here first of all the first question is guys what kind of distribution are we dealing with here so we have um 25% of all students enrolled in high school XYZ are taking algebra okay good we note that okay somebody said 0.1662.

[00:39:49]Somebody said 0.1124.

[00:39:53]Okay. Anybody with a different answer to those two?

[00:40:01]Someone said 0.1662. So when you look at this question, the first question is what kind of distribution does this model?

[00:40:10]You don't just assume. All right? You look through and try to understand what's going on. 45% of all students enrolled in high school XYZ are taking algebra.

[00:40:21]30 students are chosen at random. Oh, when you see 30 students chosen at random, first of all, I'm already taking my mind off Benoli because I know that it can never be Benoli once I'm having 30 students. 30 trials will never be a Benoli distribution. All right. Now, but does that mean it's going to be a binomial? I will tell you a reason why you have to be careful. I will tell you after this question, we're going to go into something that would blow your mind, I guess. Uh, find the probability that exactly seven students out of the 30 students chosen are taking algebra.

[00:41:00]Okay, you guys are saying it's a binary distribution. Let's confirm that. Is it binomial? Number one, 30 students. That already gives me an hint that I'm dealing with more than one trial. So my mind is already going into binomial already. But do we have uh how can we detect uh two possible outcomes?

[00:41:23]So we have 25 of all students enrolled in high school are taking algebra.

[00:41:27]Are you seeing something? 25% of all students enrolled in high school are taking. 25% are taking algebra. That means the success of getting somebody taking algebra is 25%. What is 25%? say is 25 over 100. That means just like you you know this is probability of success right probability of success for you to write a probability as 25% that means you are saying 25 out of 100 that means if for every 100 if you pick if you had 100 student 25 of them will be taking algebra that means probability and then you have to write that as obviously 0.25 25 good now 25% are taking algebra if that means in this case success will be taking algebra not taking algebra so two possible outcomes when you get into this high school XY Z if you select a student in that high school we said you could either get that student taking algebra and it's possible you also get not algebra so the two possible outcomes here is taking algebra and not algebra not taking algebra. So of course I can see that this is having two possible outcomes. So obviously I'm already leaning to binomial because I'm not seeing any reason why this should not be a binomial. Now the next question is what's going to be of course my Q is going to be 1 minus P which will be 0.75. I believe you should start mastering all these simple simple things and get very stronger in solving these questions. Once you're able to get your P, your Q is very simple. You've gotten your N. Your N is 30.

[00:43:02]All right. Your N is 30. Now find that exactly seven students out of the 30 chosen are taking algebra. That means what's going to be my random variable, guys? What's going to be my random variable? What am I going to depict my Can anybody give me that? Let me see. I want you guys to know how to do that more fluently. What's going to be my X?

[00:43:21]What am I going to say my random variable X represent?

[00:43:24]What does my random variable X represent?

[00:43:29]My honorable X can let me try and wait if I can see somebody that will say the right thing. What what will I represent my honorable X as like I mean no which saying seven? No.

[00:43:43]I'm saying what does it represent?

[00:43:44]What's the word? What does it represent?

[00:43:46]No, no, no. I don't want to I'm not saying you should give me uh what is what does it represent? Like I as you say X represents the number of students taking or GRA do you get that exactly I like unnamed unnamed without a name rep number of student taking that's the correct answer I want to hear all right X represent the number of student taking algebra so in this case now you can see that when I write probability of X now because I know X represent number of takea equals to of course the first exactly 7 seven you can see that I'm flowing in line I'm very much in line all right so in this case now you go on to use your PMF which I believe at this point now we should be more uh fluent with it n combination x that will be 30 is our hand that will be 30 combination 7 P 0.25 raised to power what? Our hex is 7 then Q 0.75 raised to power what 30 - 7 that will be 23.

[00:45:02]Can you find this quickly and tell me what you got? What's the response here?

[00:45:10]Of course, you should get 0.166 236.

[00:45:18]All right.

[00:45:21]This is how beautiful these questions are. You can literally get 100% in any question I given on all these. But you just have to be very stand to know how to detect the kind of distribution of course and even if you're able to detect the type of distribution, you must know how to solve that distribution. You must know the PMF. You can see how much we're using the PMF. So you must know the PMF and not interchange anything. know exactly what it is. Know how to apply everything. Know how to bring out your P value, your P and your Q. Know how to bring out your N. Once you do that, you are good to go. You're good to go. So guys, uh this brings me to something very important I need to expose you guys to very very important. In fact, if I if I don't talk about this, your chances of passing questions on private situation will reduce because this next thing I want to talk about is very common in exam.

[00:46:18]It's actually in your past question like I mean if you saw that question right now, if I give you that question right now, uh many of you will miss it. Almost all of you by the way like 90 9% that 1% will mean somebody that had probably researched differently.

[00:46:37]All right. But I don't want to make that attempt to try to ask you that because something you're not aware of. You are not supposed to be faulted if you miss it. So let me expose you guys to it.

[00:46:49]Guys, there's a time when you will see a question that is so perfect with binomial distribution.

[00:47:03]So let me name this subtopic binomial and poison distribution.

[00:47:12]not a very good nomenclature but just let's name that because uh you will see the connection between binary and poison in this next thing I want to say all right somebody just I just saw a comment that said why is it not geometric oh can you imagine I was not done with all the questions the previous question guys and I cleaned the okay how many of you got okay next question uh said once are fewer than Five students out of the 30 who are selected are taking algebra.

[00:47:46]Somebody was saying why is that not a geometric? Uh guys geometric is a very special distribution among every of the four of them. You should be able to spot a geometric division without any mistake because geometric is so peculiar in the sense that nothing else is like it.

[00:48:04]Geometric talks about the number of trials you have to do to get your first success. So that's the only time you're going to be saying I'm dealing with geometric when you are dealing with how many trials will you do to get your first success. All right. And I don't see anything in this question has to do with number of trials to get your first success. So everything we've been talking about so far don't have any correlation with geometric. But very soon you will see questions that has correlation with geometric. All right.

[00:48:33]Now let's try and see if we can solve the those of you that still have your data. Of course, let's try and solve the remaining two questions before I talk talk about binomial and poison distribution. So, what's provided that fewer than five students out of the 30 who are selected are taking algebra?

[00:48:47]Fewer than five.

[00:48:50]Fewer than five question are fewer than five.

[00:49:01]Probability I want you guys to be probability that fewer than five five students out of the 30.

[00:49:19]What is the probability that fewer than five students out of the 30 who are selected are taking algebra? Guys, can you attempt this? Probability that fewer than five out of the 30 students are chosen at random. What's the probability that fewer than five out of the 30 are taking algebra?

[00:49:40]That is what you what's your attempt to that Promises are fewer than five out of the 30. Can you make an attempt? Okay, just tell me what you have to do, guys. Tell me what you have to do so we can move out of this question. I'm eager to jump into some other questions.

[00:50:21]promises are fewer than five out of the 30.

[00:50:29]All right, let me just tell you what you have to do. When you get the question fewer than five, fewer than five, fewer than five it does it must be fewer than five. That means it must be lesser than five. Less than five. So we're looking for probability of xable now being less than five. And this will encompass probability of x= to0. I've told you guys that many people always make this mistake of ignoring this part. x=0 is part of it. Plus probability of x = 1 plus probability of x = 2 plus probability of x = 3 plus probability of x = 4. These are the five different probability you would have to get adding them all all up together to get the response for this. The reason for this is because they said fewer than five. So you must not include five.

[00:51:23]All right. So in this case now uh do for when p was when x was equals to zero. These are questions that are quite tedious. You have to be patient to know how to solve them. Very soon I'll be explaining to you how to use a poison and binomial table that makes the work easier. So we're going to talk about that in very soon. Not in this class obviously. I'll be sending you the tables and you'll know how to make use of them to answer questions like this.

[00:51:51]All right. So, what you have to do here is to go on to say uh you know your x being zero. Of course, you put it in the same formula and get an answer. Can you guys do for p= this first one and this second one and let me confirm that you know what you're doing. Just do for this first two then I will tell you the final answer.

[00:52:12]You will go on to practice this on your own to get to the final answer. But just do for the first two do for when x was equals to z and for when x was equals to 1. So obviously you want to be saying 30 combination 0 p which was 0.25 raised to the power of 0 then 0.75 raised to power of 30 plus 30 combination 1. That's for the next one 0.25 25 raised to power 1 and then 0.75 this success raised to power 1 failure raised to power 29 in this case that's n minus one plus on and on and do look do for these two and tell me your answer so I'm sure you know what you're doing so if you're done with the first one the very first an answer 0785 821 plus 0.17 858 plus of course let me try and tell you the answer for x = to 2 you will have 0863146 plus we have uh 026 85 5 plus we have 0.06042.

[00:53:45]Please try and practice at this stage so that you are very uh strong in these calculations. So this is this time for you to use all these answers I've given you to confirm them so that you can be very solid and fast in solving these questions. You know your exam is not just going to feature accuracy alone is also going to test your ability to solve it in record time. So the total answer the final answer should give you 0.09786.

[00:54:17]Now the next question under the same question say for you to calculate the mean and standard deviation. Guys, can you tell me quickly what's going to be the mean and standard deviation of this particular question we just solved?

[00:54:29]What's the mean?

[00:54:33]What's the mean and what's the standard deviation?

[00:54:39]Anybody?

[00:54:41]Somebody said the mean is 0.25 or name said the mean is 0.25. I told you guys this is a binary. I told you that the mean is np. Have you forgotten? I told you the variance not standation. Our variance is NPQ. You need to note all these things.

[00:54:59]Of course, uh the mean will be 30 time p which is 0.25.

[00:55:11]And that should give you an answer of what? 7.5.

[00:55:17]All right. And your variance will obviously be 30 * 0.25 25 time 0.75.

[00:55:29]That will give you uh of course for you to find the standard deviation all you have to do is to square root this. So you put all this on square root that will be the standard deviation and you should get a final answer of 2.3 717.

[00:55:47]All right, for those of you that wrote 0.25 25 as your mean. That's for Benoli.

[00:55:53]Of course, Benoli has a mean of P and a variance of PQ, but binomial is NP and variance NPQ. All right. Are you learning something?

[00:56:05]Good. So, we are good to go to the relationship between binomial and poison distribution.

[00:56:16]Quickly, if you yet to give us a thumbs up, please do that quickly. You know how important that is. That's your little way of saying thank you. Give us a thumbs up.

[00:56:28]Okay. Uh let's talk about binomial and poison distribution and the relationship between both of them.

[00:56:35]We have uh so far so good. Uh I hope you guys are flowing. Hope we don't have anybody stranded in knowledge. Hope you guys are flowing. If you have any question, you can push it in the chat box. Let's talk about it. But quickly, I want to tell you guys something very important. This is something that uh examiners try to hook students on this. They really try to do this because or whoever is not very uh sad, they cannot detect what's going on here. Now we have a scenario where you're going to see the question. The question is going to be screaming binomial on your face. This is what is going to be screaming.

[00:57:23]All right.

[00:57:24]We don't have priest in this class. I've not seen priest say something.

[00:57:29]All right. Uh the question is screaming binomial at your face. But when you look closely, when you look closely, you would see that you cannot use binomial.

[00:57:46]All right? You would notice that you're supposed to use a poison distribution.

[00:57:53]So the question you guys should ask me is when is that going to happen?

[00:57:57]Everything is going to look so much like binomial. It's going to be so much like binomial, but it's going to necessitate you applying poison. This is the moment where you have to take note of when your number of trials is very large and your probability of failure is very small.

[00:58:27]When these two things happen, always open your mind up to the use of poison distribution.

[00:58:43]So, it's my duty now to tell you uh what you are going to see. Let me give you an example.

[00:58:59]Uh this is just an example.

[00:59:03]Imagine you were told that you had 500 trials.

[00:59:10]500 n was 500 and the probability of success was 0.01.

[00:59:24]Are you seeing here that probability of success here is of course greater to one out of 100 which is 0.01 01 very low, very small.

[00:59:39]Whereas the number of trials is large.

[00:59:51]Are you guys still there?

[00:59:55]Can you still hear me? Can you indicate if you can still hear me?

[01:00:20]Okay, good.

[01:00:31]Okay, good. Okay, thank you. Thank you.

[01:00:34]Now, uh look at what I was saying. I said when you see a case where the number of trials is large and the probability of success is small this is screaming binomial when you see it but you cannot use binomial what you have is to use horizontal distribution.

[01:00:55]I'll tell you how to do that very soon.

[01:00:58]So generally there are different rule of thumb to terms that apply to this but let me tell you the truth. when you see a situation like this, you would know it. It's not always uh hard to to to detect it. Now, but there's a rule of thumb that you may want to follow and this is it. You know, we have different things being said. Some people may say, some people will say when n is greater than or equals to 50 is when you should switch to poison and of course when your p is uh less than or equals to 0.01 01 sorry 0.05 uh this rule of thumbs sometimes may be based on uh the book you are reading and whatever but what I'm going to tell you right here is for you to try it this way uh if your NP is less than or equals to five.

[01:02:00]All right.

[01:02:02]If your NP, if you can understand what I mean by NP, that means your multiplication of your number of trials and your probability of success. So in this case now you can see that NP here is 500 * 0.01 which will be five. So in this case now you should be switching you should be switching to a poison distribution. Let me start again from scratch. I don't want you guys to miss this. The reason is because if you open your past question right now you would see a question like this just like this.

[01:02:44]I believe either less than 3 years ago or there about maybe three or four years you will see exactly something like this where the question is screaming binomial. When you read it carefully, you would see that the number of trials is very high. I believe the one that came out was 3,000 number of trials. I can't remember what the probability of failure, sorry, of success was when you start seeing a 3,000 number of trials, even a 500, a 1,000, a5, a 2,000, guys.

[01:03:13]Even though you see it, so it looks so much like uh it looks so much like a binomial, but don't use binomial in that case. when your number of trial is very large and your probability of failure is small. And I told you that the rule of thumb that I will advise you to use even though you may see different uh settings in different textbooks depending on whatever they decide to use. But I would like to stick to this that your mean your the multiplication of n and p n and p your number of trial and probability of success.

[01:03:53]If your answer is five or less than five and this case happens use for his own instead of binomial.

[01:04:06]All right. So note this very much because some of you will see you will see a question and the question will be so much looking like binomial on if you have if you start doing things like one of the reasons why it's not going to even mostly work for binomial is how do you want to be even calculating with large uh you know 500 combination six and all those things it's going to be not very u very smooth that's the reason why poison fits very well. Now all these things can be explained using if we had uh if you could there's a visual way of seeing it at a scenario like this a division is always tending poison division when the number of trials start increasing and the probability of success starts reducing low it start the the binary division starts to tend to a poison diffusion guys I'm going to also give you a scenario you you guys are going to enjoy yourself if you stay tuned of course not in this as where you would see that you would see a binomial distribution. You see that cases just pay attention to what I would say next.

[01:05:15]You see those cases where uh I should have given you a question like that but I think I I was a bit soft with you guys. I did not give you that kind of question but probably let me just give you a scenario. Imagine I gave you a question like the ones where we saw now when we said 30 students that's 30 number of trials. Trials was 30 right?

[01:05:36]Imagine I now said find the probability that uh at least two students were selected.

[01:05:45]At least two student were selected. Like look at what I just said. You had 30 students. The one we just solved of AB of algebra. Now I said that at least two students.

[01:05:58]Of course you I'm sure you guys are familiar with how to solve that. That means you're supposed to solve for x greater than equals to two. All right, at least two students. Now look at this.

[01:06:11]Look at this thing right here. If you're going to solve this using all what we have been doing, you will have to Okay, I understand that this is still a bit you can find an easy way around it. But again, this is not a perfect example. Uh maybe I should say probability that at least 10 students or at least 15 students. Let me say this one because two students I can easily use the complimentary version to get the answer but let me not let me not flow with that one. Let me use at least 15 students.

[01:06:41]Now for you to solve at least 15 students it's hard for you to use complimentary here because if you try to use complimentary you will get hooked because complimentary would mean you want to be solving for one for when for zero 1 2 up till 14. All right, that's what you if you wanted to find complimentary of this one. You want to find for all do 1 minus this which will not be very convenient. You can see that you have to do 0 when probability was of x= 0= to 1 = to 2 3 4 5 6 7 9 10 11 12 13 14 that would be so tedious would it not? And imagine you wanted to go directly you want to solve from 15 15 16 17 up to 30. That would be tedious also. So in cases like that there is another approach to solve it that I can't tell you to solve questions like that now because I've not taught you that approach but if you get to that very soon we'll be talking about it.

[01:07:36]You'll be using a normal distribution.

[01:07:40]So normal distribution will work for scenarios like it will give you your answer.

[01:07:45]You don't have to use binomial all the time. So there are two cases now where we're dealing with where you would see a binomial question but you don't necessarily have to use a bin. Number one is if you're in a scenario where your what you have to solve is quite big. Maybe you're dealing with uh let's say 30 trials now and you have to solve about different probability of x being maybe from 15 up to 30.

[01:08:11]In that case now you know you're dealing with a large number of solution which is not very visible. It's not very u it doesn't stand well for you to be in an exam hall and you find parity of x= to 15 = to 14 = to 16 = to 17 18 19 20 up to 30 that would be very tedious. So in that case there is a beautiful solution by using normal division. Of course, nobody has probably talked about we've not talked about normal that will probably will start tomorrow by God's grace. So normal is very much in line when you have a situation like this. Now but the one I was trying to explain some minutes back is a situation where you use poison and that one is different from this one I just mentioned in that one when you have your number of trial being very large like 3,000 and your probability of success is low sometimes will even be 0.005 05 or 0.0 uh 1. All you have to do in that case just if you have whatever you have just go on to do uh just go on to do n * p and check up if it's less than or equals to 5 then switch to a poison distribution. Now how do you solve it using a poison distriution? That's now the next uh thing you have to get out of the way. It's so simple. But before we get into that, let me remind you of what uh a poison division entails. Poison division said the probability of x = to x is equals to what? Anybody? Can you remember what we talked about? E^ minus lambda lambda power x over what? X factorial. And we set our X here from what?

[01:10:06]From 0 1 dot dot dot. We don't have number of trials here. So we don't deal with number of trials. Don't forget horizontal division is that aspect that deals with average rate. Average rate.

[01:10:20]Good. Now you have a question that said the number of trials was 500. The probability of success one 0.01.

[01:10:31]And we're telling you to find probability that uh maybe uh let's say that x the probability of let's say you asked to find probability of x = to maybe 10.

[01:10:45]So you had this parameter n was 500 p was 0.01 find probability of x= to 10.

[01:10:54]All right. So if you wanted to use binomial normally you would have been saying 500 combination 10 uh P 0.01 raised to power 10 then Q which will be 1 - 0.01^ 01 raised to power 500 - 10. That would be 490.

[01:11:12]This is what you would have been if you dealing with binomial. We're not dealing with binomial. Poison poison. So how do you deal with this with poison? Now I'm going to ask you a quick question guys.

[01:11:25]What's the mean?

[01:11:28]What's the mean of a poison distriution?

[01:11:30]Can anybody tell me what's the mean of a poison distriution?

[01:11:40]What's the mean of a poison distribution? Anybody?

[01:11:46]Lambda. Thank you very much. What's the variance of the poison distribution?

[01:11:55]What's the variance?

[01:12:00]What's the variance of a poison?

[01:12:02]Anybody? The variance is what? lambda.

[01:12:05]Okay, so this is a special division that has the mean and the variance to be the same thing. All right. Now, what's the mean of a binomial distribution?

[01:12:20]NP.

[01:12:22]What's the variance?

[01:12:27]NPQ.

[01:12:28]So what I want to tell you right now is when you have a situation like this that it's so much like a binomial distribution but you want to use poison.

[01:12:39]In that case for you to get your lambda don't forget you want to use a poison what you need in a poison. Let's say this was a question you trying to find you had n as 500 you had p as 0.01 you want to find probability of x= to 10 and you don't want to use a you're not going to be able to use a binomial it's not it's not what you it's not adequate. you want to use a poison. So if you're going to use a poison, of course you interested in getting your lambda. Of course, you already have your X. This is your X 10. All right, that one is gotten. But the only thing left out now is your lambda if you want to use a poison. And that's what I want to expose to you. Now your lambda will be equals to NP when you're dealing with a scenario like this.

[01:13:23]All right.

[01:13:25]So our lambda when you have a sitation where it's a binomial distribution but you want to be using poison distribution to answer it your lambda there will be equals to np.

[01:13:37]So that means in a situation like this when you are when you are having your n as 500 and your p as 0.01 of course your lambda will be np that will be 500 * 0.01 01 that will be five. So what you have to do in that case is to say probability of x = 10 which was the question you wanted to solve will be equals to e rais^ - 5 then 5 to power what's your x 10 over 10 factorial. This is what you'll be having to solve in uh in the way this is what you have to solve to get your answer. So take big notes of what I just explained because this is your pathway to being able to solve any question under binomial. There are situations where they're going to give you something that will look so much like binomial or you would have to All right.

[01:15:19]Uh I want you to try and solve. Let me see what you guys are going to do.

[01:15:25]So we will be running the class very soon. I believe uh a good uh the knowledge a lot of knowledge has been passed. So let's try and solve about two more questions. A discrete random variable X has poison distribution with mean lambda.

[01:15:47]A discrete random variable as poison division with with mean of lambda.

[01:16:15]Given that probability of x = to 8 is equals to probability of x = 9.

[01:16:36]Determine the value of Now this is a question for you to solve quickly.

[01:17:05]So you are told that in this random variable X in this random variable X I should have haded X there has a poison distribution with mean lambda given that probability of X = to 8 = to probability of X= to 9.

[01:17:29]Determine the value of probability of what you have here X between you know you can see here we have less than or equals to here we have less than. So, can you make uh can you make an attempt to this quickly?

[01:18:06]Anybody with an answer? What's your attempt to this? What do you feel you have to do?

[01:18:12]Uh you you told to find the value of priority of x between 4 and 10. 10 included, four not included.

[01:18:23]All right. What you have to do here is for you to number one try to identify what you have to bring out. This means you are getting.

[01:18:35]So obviously these are the values you want to be bringing out. These are the values you want to be bringing out.

[01:18:42]Now but this is now but this is just a a a Can you indicate if you can still hear me please?

[01:19:09]Can you still hear me there?

[01:19:16]Okay. Looks like the network is misbehaving.

[01:19:26]All right. The way to solve this is you have to solve this question and what you have to do is to uh find a way of getting your lambda through this expression that was given. This means if you're told that probability of x= to a equals to probability of x = to 9.

[01:19:49]Obviously what you have to do here is to say e to power minus lambda which you know lambda then lambda raised to power x in this case 8 over 8 factorial is equals to e^ minus lambda lambda^ 9 over what 9 factorial.

[01:20:10]So if you look at this this is what you have to solve to get your lambda because you're not given lambda here. This is a very trick question, right? You have to be able to get your value of lambda yourself. And the way to get it is by solving this expression, which is what I just uh dissolved into this. So, of course, since this is a poison division, you know this an expression for this probability whereas this is also an expression for this other probability.

[01:20:37]And you were told that they equals to each other. So, mathematically, I'm sure you know this will cancel this.

[01:20:43]I'm sure you know that.

[01:20:46]All right. And we have y raised to sorry lambda raised to power 8.

[01:20:54]If you go want to cross multiply you have * 9 factorial equals to lambda^ 9 * 8 factorial. Of course you know this same thing as lambda * sorry lambda^ 8 * 9 * 8 factorial equals to lambda^ 9 * 8 factorial. 8 factorial we cancel 8 factorial. You are left with lambda rais^ 8 * 9 = to lambda raised to power 9.

[01:21:22]You go on to say let's divide both side by uh let's side by lambda^ 8 which is very possible. Divide both side by lambda^ 8 you left with 9 = to lambda. All right because this will be lambda* lambda* in n places lambda eight places lambda raised to power this will cancel this you can use the law of indices which will tell you that this will be lambda^ 9 - 8 which is still going to be lambda so your lambda here is 9. So the major thing here is for you to get your value for lambda and getting lambda as n the deal breaker which can now lead you to go on to find theity that you have to find here which as I said already it will include 5 6 7 8 9 and 10. Four will not be included because the equals to there doesn't cover it. So guys uh this is where I'm going to leave you guys.

[01:22:21]we wouldn't be able to continue for today. And you guys should make sure you go out there uh this video will be available uh tomorrow probably by 11 a.m. So go out there and immerse yourself in lots and lots of questions.

[01:22:36]We cannot solve all the questions I have. I have lots of questions but we'll keep solving as many as possible and um with the time we are permitted to have for this. So uh guys take good care of yourself. Take good care of yourself.

[01:22:53]Read very well. Read solid. Care well for all that is ahead of you. Uh tomorrow by God's grace we're getting into normal distribution which is very vital for anybody uh to really be a part of. Thank you guys. Uh have a good night. Enjoy yourself. Bye-bye.