 Hello, in this video we will talk about something that will make your life much easier and that would be binomial probability distributions. So binomial probability distribution is a type of discrete probability distribution and we'll talk now about what constitutes as being binomial. So binomial probability distribution arises from a fixed number of independent what we call Bernoulli trials. So in other words the following four requirements must be met. So the procedure has a fixed number of trials. The trials are independent that means the probability of the first trial or the outcome of the first trial does not impact the probability of any of the following trials and so forth. Each trial only has two outcomes, so two possible outcomes one of which is called a success the other which is called a failure, either it happens or it doesn't. The probability of a success in one trial is represented by the little letter P and the probability of a failure is represented by Q. Together P plus Q add up to one because remember you have only two outcomes and the outcomes total on a probability distribution the probabilities must add to one. So therefore we know that Q is always one minus P. So some notation we'll be working with N is the number of trials, X is the number of successes we're looking for, P is the probability of a success, Q is the probability of a failure and P of X represents the probability of getting a certain number of successes among the trials. So be sure that X and P both refer to the same category called a success for instance if you're flipping a coin and you're looking at the coin landing on heads a success would be the coin landing on heads that's the behavior you're looking for. So you want P your probability to correspond to the probability of getting heads. One other thing to look out for is that a success is not always necessarily a good thing like when you're talking about diseases and survival rate or people passing away from a disease a success could be deemed as someone dying from a specific disease or condition. So it's the behavior we're looking at it's not always a positive thing it could be giving a test trial of some drugs to people then a success could be the person gets a headache. So it's whatever is being observed it's not always a positive thing. So now with the following result in the binomial probability distribution or a binomial experiment whatever you want to call it if so we will identify N, P and Q. So when an adult is randomly selected there is a .85 probability that this person knows what Twitter is. We want to know the probability that exactly three of five randomly selected adults know of Twitter. So we need to run through the four conditions or the four requirements. Condition one is there a fixed number of trials yes it says five randomly selected adults so there's five trials so that's good. Two are the trials independent yes the trials are independent you go to the first person you ask them if they know what Twitter is they give you the response and you just move right along to the second person they are independent of each other alright three. Our third requirement is each trial has two possible outcomes called a success and failure so what would be deemed a success here well it'd be knowing what Twitter is and what would be the failure not knowing what Twitter is so not knowing Twitter. And then our fourth requirement was that the probability of a success in one trials P so is there a probability of knowing what Twitter is well when an adult is randomly selected there is a .85 chance that the person knows what Twitter is so yes we know the probability of a success the probability an adult knows what Twitter is so yes the experiment is binomial. Talk about why this is important momentarily. What about the following experiment five cards are randomly selected from a 52 card deck without replacement and the number of diamond cards selected is recorded so here we have a fixed number of trials five cards so my conditions we have five trials second condition independent so I go through and on my first trial I pick out a card but I don't replace it it says without replacement so what does that mean for the second trial well it means there's now one less card in the deck and what does that mean well it means it's going to change the probability of your next card selection because now you only have out of 51 cards instead of out of 52 so unfortunately the trials are not independent because of the word or the key words without replacement without replacement means not independent so immediately this experiments disqualified from being a binomial probability distribution so not binomial sorry go find another club to join so to kind of scaffold us into the binomial probability distribution formula I want us to do some probability calculations by hand so on a multiple choice test that would be choices with a b c or d as the answers there are three questions a student randomly selects answers for each of the three questions find the probability the student chooses one correct answer so I'm going to let c equal correct and let w equal wrong so my ultimate goal is to find out what is the probability that the student gets one correct answer out of the three questions so let's think about all the ways the student could get one answer correct I would have to calculate okay first answers correct the second two are wrong I could get only the middle question correct I could get only the last question correct those are there's three ways you could get exactly one correct answer and I'm going to calculate the probabilities of each of these that's why I put the equal signs there it's getting ambitious so what's going to happen here is I need to calculate okay the probability I get the first question correct times the probability I get the second question wrong times the probability I get the third question wrong so this is literally going to be okay what's the probability you get the first question correct out of four answer choices there's only one that is correct so one fourth times what is the probability you get the second question wrong there's three wrong answers out of four so you have times three out of four and the same goes for the third question three out of four so that's going to give me when I multiply everything together I get point one four one okay so that's one outcomes answer point one four one well let's do wrong correct wrong well what is the probability of let me write this out what is the probability of getting the first question wrong and that's three out of four second question correct times one out of four and then third question wrong three out of four again it's the same fractions being multiplied together in a different order so still point one four one all right let's do wrong wrong correct now and you might kind of know where I'm heading here it's literally three over four times three over four times one over four you're going to get point one four one you do three times three you get nine four times four times four to get 64 nine over 64 is point one four one the same probability again so even though my correct answer was a different one of the questions in each of these possible scenarios I still got the same probabilities so if I add up these three three decimals or these three probabilities I get point point four two three that is the probability you get exactly one answer correct out of the three now you want to take this into account because literally all I could have had done was know that there's three possible outcomes in terms of getting one of the questions correct and all I had to do was calculate one of the probabilities because once you know one of the probabilities times it by three because there's three different ways you can get one correct answer and that would give you the answer as well so there are some shortcuts there but that is just one possible scenario is getting one answer correct you could get no answers correct you could get two answers correct you could get three answers correct if I was building a probability distribution I would have to compute the probabilities for four rows and that's very exhausting so we have to have some sort of shortcut or remedy for that sort of tedious task so that's where we're about to learn so if an experiment's binomial I have good news for you if an experiment's binomial then that means we can calculate the probability of a certain number of successes or outcomes occurring using a formula or using google sheets so what the formula states is the probability of a certain number x successes occurring in n trials is the following so literally you take the number of successes you have or you want and you multiply the probability of a success together that many times so for instance in the previous question I wanted one correct answer so I had one fourth appear in each of my calculations and then you multiply by the probability of a failure to how many times or how many failures you'll have so for instance in the previous exercise the probability of a failure was three fourths I had two wrong questions in each of my possible scenarios so I multiply three fourths twice in terms of my probabilities and then to take in the account the fact that you can get one correct answer or whatever you're dealing with in various ways for instance if you have three questions the number of ways to get only one correct is given using the combination formula three choose one that's what this n factorial expansion is it's the combination formula so that's where the probability binomial probability formula comes from so you're taking your probability of a success multiplying it by how many successes you have you have probability of a failure multiply that failure probability by how many failures you have so if your failure probability is three fourths you do three fourths times three fourths if you're interested in two failures and then you have your combination formula to represent that certain number of successes kind of occur a certain number of ways so it's kind of a cool formula and it's got a really nice background to it but we need to practice a little bit on how to use it so just to recall n is the number of trials x is the number of successes you're looking at p is the probability of a success in a single trial and q is the probability of a failure in a single trial q equals one minus p so there's my explanation of the the probability formula for the binomial experiments like I said we'll use google sheets to do the calculation but you should know kind of where the formula comes from otherwise you won't appreciate it as much so we will use google sheets i'll show you that in a moment so on a multiple choice test there are three questions a student randomly selects answers for each of the three questions find the probability the student chooses one correct answer let's do this binomial style so the experiment is binomial so feel free to look at your notes or look at the previous video or the beginning of this video and make sure you know why so we have a certain number of trials we have three questions and we have a success a success is getting an answer correct or getting question correct i should say and what is my probability of a success what's p what's the probability of getting a question correct well there's a one in four chance of getting a single question correct that's a point two five probability so literally all you need to know when you go through and do google sheets is i'll make a little list over here when you go through and do google sheets you need to know how many trials you have you need to know the probability of a success this may not be the order we input them but i just need to list them and you need to have the number of successes you're looking at you're looking at exactly one question correct so i'm looking for the probability that the number of questions correct is equal to one so let's go to google sheets so we will go to the compute tab and we are going to focus our attention on the binomial region so we input in first our number of trials our number of trials is three the probability of getting a question correct is point two five that's our p and then lower bound and upper bound if you're only interested in one success or one correct answer you write from one to one that's saying i'm only interested in the outcome when there's one success and there's one question correct so when you have exact you use the same number for both the lower and upper bound so you get about point four two two so this is more precise than our previous calculation we get point four two two because we don't have to worry about any crazy rounding error in our mid calculations so this formula is really nice gives us more of an exact answer let's continue the practice using the binomial formula for calculating probabilities in a clinical test of a drug 81.3% of the subjects treated with 10 milligrams of a certain drug experienced headaches five subjects were randomly tested five subjects and then 81.3% experienced a headache find the probability exactly three subjects experience headaches so a success is getting a headache that's the behavior of the outcome we're looking at i told you it's not always something positive so success is getting a headache what is the probability of someone getting a headache it's 81.3% or 0.813 so find the probability exactly three experience a headache so google sheets the number of trials is five there's five subjects the probability of getting a headache is 0.813 and then we're looking at exactly three so that means lower bound and upper bound will both be three we'll go to google sheets in a minute next find the probability that more than one subject will experience headaches so the probability that the number of people getting headaches will be greater than one so n equals five p equals 0.813 and we're looking for x is greater than one is it reasonable to expect that more than one subject will experience a headache we'll analyze that after we do our calculation so when went when i go to google sheets when x equals three for part a i'm going to type three for both my lower and upper bound when x is greater than one my lower bound will be two just think about numbers that are greater than one it would be two and my upper bound would be five because you can't have any more than five there's only five trials so you can't have more than five people with a headache so let's go to google sheets and number of trials is five probability of someone getting a headache is 0.813 for part a we said exactly three so point one eight eight point one eight eight and then for part b we said we're looking for greater than one or more than one so our lower bound would be two you can't do one because more than does not include one and it goes all the way to five get 0.995 0.995 so we got 0.188 and then we got 0.995 so is it reasonable to expect more than one subject will experience headaches look at the probability from b i say yes that's because the probability x is greater than one that more than one person will experience a headache is 0.995 that's an introduction to binomial probability distributions and how to use the formula for their calculations thanks for watching