Understanding Binomial Distribution Concepts

hello everyone welcome to philos academy today i'm going to take you through binomial distribution so without wasting much time let's start first let's look at the meaning of binomial experiment and binomial distribution now we say a binomial experiment is a probability experiment that satisfies the following four requirements so the first requirement is that there must be a fixed number of trials so when you toss a point five times that five is the number of trials we should have a fixed number of trials and secondly two outcomes that is success and failure are possible on each child so on each trial two outcomes are possible either success or a failure and also the outcome of hr must be independent of one another this means that the outcome of the second trial shouldn't depend on the first one or the first one shouldn't depend on the second one they should be independent of one another and also the probability of a success denoted by p must remain the same for each trial okay now if the probability of success in the first trial is 20 that is 0.2 it should be the same in the second trial it should be the same in the third one it should be the same in the fourth one going okay the outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution so let's look at the formula we're going to use in this talk now in binomial distribution the probability of an event x occurring should be equal to n combination x times p exponent x then times 1 minus p exponent n minus x this formula can also be written as n combination x p exponent x then 1 minus p exponent n minus x one thing i want you to know is that you can replace q with one minus b okay so you can write the formula as n combination x b exponent x then q exponent n minus x what matters is you should know that q is equal to 1 minus p since you know that n combination x is equal to n factorial divided by n minus x factorial times x factorial then the formula can also be written as n factorial divided by n minus x factorial x factorial p exponent x then one minus p or q exponent n minus x you can also write the formula like this okay now n stands for the number of trials okay and then p stands for the probability of success in a single trial okay and then one minus p which is the same as q represents the probability of failure in a single trial okay and then x is the number of successes okay so this is the formula we're using binomial distribution so i'm going to write mine this way and combination x p exponent x then k exponent n minus x please remember q is the same as one minus p and then have written n combination x here still the same as this okay so but i'll be writing mine in this form so you can choose to write them in any form but i'll be writing mine like this okay now before i proceed in binomial distribution to find the mean the mean is referred to as expected value of what of x the mean is equal to n times p number of trials times the probability of success in a single triad okay so that will give you the mean and also to find the variance that is variance of x the formula is n times p times q or times 1 minus p so that's how we find the variance and then to find the standard deviation that is sd of x it should be the square root of the answer you got for the variance so square root of n times p times q that would be the standard deviation so let's take some questions and see how to do this properly now question one says a point is tossed five times find the probability of getting exactly two tails now to solve this question first what is the binomial formula the formula says that probability of x equals x should be equal to n combination x isn't it p exponent x times q exponent n minus x now from the question the question says a point is tossed five times it means that the question has given you n to be wrought to be 5 find the probability of getting exactly two tails every question will give you the n and the p and ask you to find the x okay now with the help of the p you can know the kill yourself but yeah like the case that when you look at this question it looks like the p is not in but it is in let's analyze this you said a coin is paused five times now you ask yourself when i toss a coin once okay when i toss a coin once there are two possible outcomes isn't it so since there are two possible outcomes the probability of getting a tail in a single tossing should be number of tails in a single torsion you know in a single tossing there are two possible outcomes tail or head so probability of getting a tail in a single torsion should be number of tails in a single thousand divided by number of outcomes in a single torso which is two so we've gotten p to be one over two and then together q the q is one minus p isn't it which is equal to one minus our p was one over two so that is going to give us 1 over 2 also so we can now substitute in the formula but the question says find the probability of getting exactly two tails so to find the probability of getting exactly two that x is exactly two what we say exactly means that it is equal to two okay will give us what is our end according to the question it is five so five combination x is two so combination two then rp is one over two which is the same as zero point five okay so 1 over 2 we can write it 0.5 1 over 2 exponents 2 times our q is also 1 over 2. now n is 5 according to the question isn't it and then our x is 2. so we're going to have now when you put 5 combination 2 on your calculator to get it first enter five okay then click on shift and then the division sign you get a combination now i get a combination it comes as a symbol c then you click on two so five combination 2 give you 10 times 1 over 2 exponent 2 will give you 1 over 4 that is 1 squared divided by 2 squared okay you can use your calculator and then five minus two is three isn't it one over two exponent three should give you one over eight now when you simplify this you get what five divided by what 16. so in decimals it is 0.3125 leave your answer in four decimal places okay so in decimal it is 0.3125 so this is the answer to the question now if the question had asked you that you should find the mean okay the mean which is expected value of x should be equal to what n times p and what is our n our n is five and our p is what one over two so five times one over two should give us what two point five so the mean is what is two point five for the variance that's variance of x it should be n times b times q isn't it so our n is what five isn't it and then p is one over two and q is one over two so five times one over two times one over two should give us what one point two five so this is the variance so the variance is what 1.25 if you want to find a standard deviation then a standard deviation should be what square root of what you got for the variance so square root of 1.25 so square root of 1.25 will give you us one point so this is the mean 2.5 and the variance is 1.25 and the standard deviation is 1.118 the question only asks for the probability of getting exactly 2 too so this is the answer 0.3125 so let's proceed to the next question question 2 says it is known that 37 of inhabitants of a community favor a political party pa so this is the name of that political party now a random sample of 30 inhabitants was selected from the community and each person was asked if he or she will vote for pa party in an impending election what is the probability that hey no one will vote for pa party so let's see how to solve this now for this question rp is 37 so you have to convert it towards decimals or fraction okay so 37 divided by 100 will give us 0.37 okay then our q is going to be 1 minus p which is 0.37 so 1 minus 0.37 give us 0.63 okay so that's our kill and then from the question they said a random sample of 30 so this 30 is the n the sample size okay so that's now to solve question a question a says what is the probability that no one will vote for party it means that we should find the probability that x will be exactly zero no one will vote for party a it means x will be out exactly zero so to solve this what is our n our n is what 30 so we say 30 combination x which is zero then rp is what rp is 0.37 so we say 0.37 exponent x which is 0 times q is 0.63 so 0.63 exponent 30 minus 0 that is n minus x okay so let's simplify this you can put this on your calculator directly okay now 30 combination zero will give you one times any number exponent zero also give you one okay times zero point three exponent 30 minus 0 is k80 so if you simplify this with your calculator you'll get 0.0000955 so that would be the answer okay now let's look at question b question b says we should find the probability that exactly two persons will vote for ba party so we want to find the probability that x is exactly 2. so that will be what n combination 2 that's n combination x then p exponent 2 times q which is 0.63 exponents 30 minus 2 now you can put this direction on your calculator to do that you input 30 then you click on shift and the division key and that will give you the combination symbol then you enter 2 then you bring your multiplication sign times open a bracket and then enter 0.37 in the bracket close it and square it then bring times enter 0.63 in the bracket and then exponent 30 minus 2 so just click equal to sign you get answer straight forward so 0.00014 so that's the answer to your question b okay now question c says you should find a probability that at most two persons will vote for pa pattern so question c says at most two persons now when we say at most at most means that maximum should be that value so i would say at most two it means that maximum should be two it shouldn't be more than two okay so at most two means that x is less than or equal to two it means that it should either be less than two or equal to two but not more than two that's the mean of at most two okay so if we say x is less than or equal to 2 then it means that it's either x is exactly zero or x is exactly one or x is exactly true that is the meaning it's either that x is exactly zero or x is exactly one or existing do you get it so if i say i go to the studio at most twice in a year it means that it's either i don't go at all or i go just once in a year or i go there twice in a year but not more than twice that is the meaning okay so to solve this we are going to use the binomial formula to find the probability that x is exactly zero okay then you use that same formula to find the probability that x is exactly one use that same formula to find the probability that x is exactly 2. so to do that we can say what n combination was x which is 0 then p which is 0.37 0 times 1 minus p which is 0.63 exponents 30 minus 0 then plus what you've written here is just for probability that x is exactly zero so you find for this as well 30 combination one zero point three seven exponent one then zero point six three exponent 30 minus 1. now this thing that the probability of success remains the same for this and this again 30 combination 2 0.37 exponents 2 then 0.63 exponent 30 minus 2. so when you put this on your calculator you're going to get zero zero 0.0000 nine five five plus when you solve the second one also you're going to get zero point zero zero zero zero two plus the last one is going to be 0.00014 so when you add the three of them together you're going to get zero 0.0001609 five five so you can approximate it to zero point zero zero zero one six so that is the answer to question c so let's look at question d now question this is what is the probability that at least at least three persons will vote for pa party now when we say at least three it means from three upwards okay at least many the minimum should be three doesn't mean so when we say at least three means that x is greater than three or equal to three okay at least three it means that three or more so greater than or equal to three okay now to find at least three is going to be probability that x is exactly three because we say three or more right or probability that x is exactly 4 or probability that x is exactly 5 you do that up to n okay up to probability that x is exactly 30. but it's not going to be easy to find everything out okay okay so for that matter you need to know another method of doing this now to understand the second method of doing this there's something i want you to know now just like how you learn words and opposites that the opposite of good is bad the opposite of open is closed and the lies now i want to show you some ways and opposite here and i'll help you to solve this question in a simple way okay now and it goes like this look at the sign less than or equal to sign each opposite is greater than please don't include the equal to in the opposite because it's already included here so the opposite of greater than sign should be less than or equal to okay and also the opposite of greater than or equal to should be less than and also the opposite of less than is greater than or equal to it goes in that manner also the equal to sign its opposite is not equal to okay so if you notice that will help you to get a simple method to solve this question so it goes like this probability that x is greater than or equal to three should be equal to one minus its opposite so what is the opposite of greater than or equal to it is less than isn't it so it should be one minus probability that x is less than three if you're able to write this you are gone so this will give us one minus so let's break this down x less than three means from zero to two okay less than three not less than or equal to three but less than three okay so it means you open a bracket probability that x is exactly zero plus probability that x is exactly one plus probability that x is exactly two so that will give us one minus into bracket now from the answer to question c we got a probability that x is actually zero to be 0.000 zero zero zero zero nine five five so there's no need to solve anything you can just take it from there okay plus probability that x is only one is zero point zero zero zero zero two plus probability that x is exactly 2 is 0.00014 so this will give us 1 minus when you add them together you get 0.0001 so 1 minus 0.00016 will give you what 0.99984 so this is the answer to question d okay 0.99984 now in case your question asks you to find the mean variance and standard deviation to get the mean the mean which is expected value of x will be equal to n times p square n is 13 and our p is 0.37 so 30 times 0.37 will give us 11.1 okay and then the variance will be n times p which is 0.37 times q which is 0.63 so when you multiply it you get 6.993 and then for the standard deviation the standard deviation will be the square root of the variance which is 6.993 okay so you get 2.644 so that's the answer to this so let's look at another question question 3 says a multiple choice test consists of 15 questions so for this question it means that n is 15 each with 5 possible answers of which only one is correct so it means that the probability of getting one correct should be one over five because you have five possible answers and only one is correct so the probability of getting one correct should be what one over five okay if it contains four possible answers then probability of getting one correct should be worth one over four but this has five possible answers so the probability of getting one correct should what one overhead five okay so we know our n and we know our p now let's proceed suppose one of the students taking a test answers the questions by guessing what is the probability that he answers at most three questions correctly now when we say at most at most means the maximum should work three isn't it so we want to find the probability that x is less than or equal to three so that will give us probability that x is exactly zero or x is exactly one or x is exactly two or x is exactly three because they say at most three mini maximum should be three okay then let's find the probability that x is exactly zero now our n is 15 isn't it so that should be 15 combination zero and then probability of success is one over five so one over five exponent zero and then probability of failure should be cubed which is one minus one over five which will give us four over five so probability of failure is four over five so four over five exponent 15 minus zero then plus then we do that for probability that x is exactly one as well so we're going to have 15 combination one one over five exponent one times four over five exponent fifteen minus one plus fifteen combination two times one over five exponent two times 4 over 5 exponent 15 minus 2 plus 15 combination 3 1 over 5 exponent 3 times what 4 over 5 exponent 15 minus 3 we are following the binomial formula okay so when you put this first one on your calculator you're going to get 0.03 five one eight plus when you put those also on your calculator you get zero point one three one nine four plus when you put the source on your calculator you get zero 0.2309 plus when you put the last one on your calculator you get 0.25014 so when you add the four of them together you're going to get zero 0.64816 so this is the answer to question 3. in case the question asks you to find the mean variance and standard deviation to get a mean that's expected value of x the mean should be number of files which is 15 times probability of success which is one over five so that will give you three so the mean is three okay if you want to find the variance variance should be what number of channels times probability of success times probability of failure which is four over five so when you simplify these two you get two point four then to get a standard deviation the standard deviation will be square root of the variance so square root of two point four will give you one point five four nine two okay the answer to this question at most three questions okay is 0.64816 so let's look at another question question 4 6 a report from the secretary of health and human services stated that 70 percent of single vehicle traffic fatalities that are occurred at night on weekends involve an intoxicated driver if a sample of hitting single vehicle traffic fatalities our care at night on a weekend is selected find the probability that exactly 12 involve a driver who is intoxicated now to solve this the probability of success okay is 70 when you take 70 divided by 100 you get 0.7 or 7 over 10 okay then probability of failure which is killed should be 1 minus 0.7 which gives you 0.3 and then the number of trials they say if a sample of 15 so the sample of 15 is the end okay so our n is what 15 so to solve this probability that exactly 12 involve a driver who is intoxicated so the subject matter is intoxicated that's what i'll be was n combination x so our n is what 15 combination x which is 12. okay remember n combination x can also be written as 15 and write it this way combination 12 okay or you can write it this way also in form of vector then p exponent x so our p is 0.7 exponent x which is 12 times r q is 0.3 isn't it exponent n minus x now when i take 15 combination 12 you'll get 455 times now 0.7 exponent 12 should give you approximately 0.01384 times now fifteen minus twelve is three isn't it zero point three exponentially should give you zero point zero two seven so when you multiply the three of them you get 0.17004 so that is the answer to this question as well now it's because i've approximated some of the values i'm getting this if you had put this part on your calculator straight forward maybe you'd have gotten something slightly different from the it could be 0.17002 okay that doesn't make your answer wrong okay the answer is still correct okay so in case the question asks you to find the mean you just take the number of charge times probability of success that is 15 times 0.7 you get 10.5 okay if you want to find a variance then it should be 15 times b which is 0.7 times q which is 0.3 and that will give you 3.15 to find the standard deviation you take the square root of what you got for the variance okay so square root of 3.15 will give you 1.775 so that is the answer so let's take another question question 5 says a die is rolled 480 times so solution our n is 480 so meaning number of trials is 480. find the mean variance and standard deviation of the number of trees that will be ruled now when you look at this question you do not have p isn't it but for this particular question you can get a p yourself to get a p you ask yourself when a roller die once what is the probability of getting a 3 in a single rolling okay so the probability of getting a three in a single ruling should be one over six the reason is there is only one three in a single ruling isn't it and there are six possible outcomes in a single rolling of a diet so number of threes divided by what number of possible outcomes which is one over six okay so p is one over six so to answer the question the mean which is expected value of x should be equal to n times p which is 480 times one over six four hundred and eighty times one over six will give you eighty okay then to find the variance this is how write a variance this symbol is called sigma okay it's a greek letter it's called sigma it's a small letter sigma this is a capital letter sigma and this is a small letter one okay so sigma square is a symbol for representative variance so variance of x should be equal to n times p times one minus p or n times p times q okay so that will give us 480 times one over six times what is our q our q is 1 minus 1 over 6 which is 5 over 6 okay so [Music] 5 over 6 now 480 times 1 times 5 will give you 2 400 divided by this is over 1 1 times 6 times 6 will give you 36 you can use your calculator okay when you simplify this you get 200 over three okay if you are leaving enzyme mixed fraction then it should be 66 whole number two over three if you are leaving answer in decimal then it should be 66.667 now to find the standard deviation now the standard deviation the symbol is sigma without a square okay so ld of x should be equal to square root of what you got for the variance okay square root of n times p times q so square root of 66.667 will give you 8.165 so that's the final answer to this so let's proceed to the next question question 6 says a large retailer purchases a certain kind of product from a manufacturer the manufacturer indicates that the defective rate of your product is three percent in a shipment the inspector of the retailer randomly pick 20 items of the product from a shipment what is the probability that there will a be three defective items now from this question the probability of success is three percent that's three over hundred and i'll give a zero point zero three okay and the number of trials is and then probability of failure which is q should be 1 minus success which is 0.03 so 1 minus 0.03 will give us 0.97 now to solve question a it says if you find the probability that there will be three defective items so we are finding probability that x is exactly three so that will give us n combination x so our n is what 20 so 20 combination 3 times rp is 0.03 so 0.03 exponent x which is 3 times our q is 0.97 exponent n minus x and our n is 20 minus our x is 3 isn't it so 20 combination 3 if you put this on your calculator you get 1140 times 0.03 exponentially should give you 0.000027 times now 20 minus 3 is 17 0.97 exponents 17 will give you 0.5958 so when you multiply the three of them you get zero points 0.01834 so that's the answer to question a now question b says what is the probability that there will not be more than two defective items meaning there won't be more than two so when we say there won't be more than two it means that the maximum should be two but not more than two okay so in other words this means at most two okay not more than two means at most two okay so probability that x is less than or equal to two should be probability that x is exactly zero or x is exactly one or x is exactly two so let's find the probability that x is exactly zero that would be n which is 20 combination x which is zero times zero point zero three exponent 0 times 0.97 exponent 20 minus 0 plus then we do that for the second one also 20 combination 1 p is 0.03 exponent 1 which is x times 0.97 exponent 20 minus 1 then plus 20 combination 2 times 0.03 exponential times 0.97 exponent 20 minus 2. so let's simplify this you can input this on your calculator straight forward okay now when you solve the first one you're going to get 0.54379 plus when you solve the second one also you get 0.33637 plus then the table will give you 0.09 eight instead okay so when you add the three of them together you're going to get zero points nine seven eight nine nine so that's the answer to question b now in case the question asks you to find the mean variance and standard deviation for the mean the mean is going to be n times p that is 20 times 0.03 and that will give you zero 0.6 okay for the variance it should be n which is 20 times p which is 0.03 times q which is 0.97 and that will give you 0.582 and then to find the standard deviation the standard deviation should be square root of the variance and square root of 0.582 should give you 0.763 so let's proceed to the next question question 76 it is known that all items produced by a certain machine will be defective with probability of 0.1 independently of each other what is the probability that in a sample of three items at most one will be defective now from the question rp is 0.1 our q is going to be 1 minus p which is 1 minus 0.1 and that will give us 0.9 okay and then n is 3. and the question says we should find the probability at most one will be defective so we say at most one meaning the maximum should be one so it means that x is less than or equal to one okay and that will give us probability that x is exactly zero or x is exactly one so let's find the probability that x is exactly zero and probability that x is a valley one then we add them together so that will give us three combination zero times probability of success is zero point one isn't it exponent zero times now probability of failure is 0.9 exponent 3 minus 0 plus then we take the second one also three combination one rp is 0.1 isn't it so 0.1 exponent 1 and an rq is 0.9 isn't it so 0.9 exponent 3 minus 1. so 3 combination 0 will give you 1 times 0.1 exponents there is also one times zero point nine exponents three minus zero is three okay plus now three combination one will give you three times zero point one exponent will give you 0.1 okay times 0.9 exponent 3 minus 1 is 2 okay now when you simplify 1 times 1 times 0.9 exponent 3 you get 0.729 okay plus 3 times 0.1 times 0.9 exponent 2 will give us 0.243 so when you add the two of them together 0.729 plus 0.243 you get 0.972 so the correct answer for question 7 is 0.972 okay now in case the question asks you to find the mean expected value of x it should be number of child which is three times the probability of success which is 0.1 so 3 times 0.1 gives you 0.3 so that's the mean and then the variance the variance should be number of childs times probability of success times probability of failure so three times zero point one times zero point slice div zero point two seven and then the standard deviation should be the square root of the variance so square root of zero point two seven is zero 0.52 so let's proceed to the next question question 8 is suppose the probability that a colored freshman will graduate is 0.6 3 college freshmen are randomly selected what is the probability that at most two of these students will graduate so to solve this probability that one person will graduate is 0.6 isn't it and the probability that one person will not graduate which skill should be 1 minus 0.6 which is 0.4 but the question says three college freshmen are randomly selected so n is what is three and also the question is saying that we should find the probability that at most two of these students will graduate so what we say at most so it means that x is less than or equal to two which will give us probability that x is exactly zero or x is exactly one or x is exactly two okay so let's find a probability that x is exactly zero and that should be what n combination x which is three combination zero then p exponent x times q exponent n minus x plus n combination 1 this time around times p which is 0.6 exponent 1 times k which is 0.4 exponent 3 minus 1 plus n combination 2 p exponents 2 times 0.4 exponent 3 minus 2. so you can put all this on your calculator you get 0.064 okay plus so when i simplify this also you get 0.288 [Music] plus then when you simplify the third one you get 0.432 so add the three results together when you add them you get 0.784 so the correct answer to question 8 is 0.784 in case the question asks you to find the mean variance and standard deviation the mean is going to be n which is 3 times p which is 0.6 so 3 times 0.6 will give you 1.8 okay to find the variance the variance should be 3 times 0.6 times the probability of failure which is 0.4 so 3 times 0.6 times 0.4 will give you 0.72 okay and then to find the standard deviation it should be the square root of the variance so square root of 0.72 gives 0.849 now this is another style of giving you a binomial question now look at this the question says the random variable x has binomial distribution b into bracket 40 comma 0.2 determine each of the following a probability that is exactly five going okay now the nature of this question is different from the previous ones that i have solved okay but what i want you to know is that whenever you have this type of question what you should know is that the first argument here is the number of childs which is 40 and then the second argument is 0.2 should be the probability of success so now you know that n and then you know the p so you can easily find the q as what one minus 0.2 which will give you 0.8 so with this known and we p also known and q known you can solve a b c d e and f okay so i want you to try this so i'm going to give you the answers so that when you try then you are correct you know now for question a i think you get 0.08541 and then for question b you get 0.95676 then for question c you're going to get 0.1607 and then for question d you're going to get 0.24881 and then question e you get 0.02846 and then for question f i won't tell you the answer try as much as possible to get an answer but make sure you work with a to e properly to check if your answer is the same as mine so i'll end the binomial distribution here for now in our next video we are going to look at the poisson distribution so if you haven't subscribed to the channel yet please do also subscribe so that whenever i post more important videos like this you can always have access to it easily thank you [Music] [Applause] [Music] [Applause]

hello everyone welcome to philos academy today i&#39;m going to take you through binomial distribution so without wasting much time let&#39;s start first let&#39;s look at the meaning of binomial experiment and binomial distribution now we say a binomial experiment is a probability experiment that satisfies the following four requirements so the first requirement is that there must be a fixed number of trials so when you toss a point five times that five is the number of trials we should have a fixed number of trials and secondly two outcomes that is success and failure are possible on each child so on each trial two outcomes are possible either success or a failure and also the outcome of hr must be independent of one another this means that the outcome of the second trial shouldn&#39;t depend on the first one or the first one shouldn&#39;t depend on the second one they should be independent of one another and also the probability of a success denoted by p must remain the same for each trial okay now if the probability of success in the first trial is 20 that is 0.2 it should be the same in the second trial it should be the same in the third one it should be the same in the fourth one going okay the outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution so let&#39;s look at the formula we&#39;re going to use in this talk now in binomial distribution the probability of an event x occurring should be equal to n combination x times p exponent x then times 1 minus p exponent n minus x this formula can also be written as n combination x p exponent x then 1 minus p exponent n minus x one thing i want you to know is that you can replace q with one minus b okay so you can write the formula as n combination x b exponent x then q exponent n minus x what matters is you should know that q is equal to 1 minus p since you know that n combination x is equal to n factorial divided by n minus x factorial times x factorial then the formula can also be written as n factorial divided by n minus x factorial x factorial p exponent x then one minus p or q exponent n minus x you can also write the formula like this okay now n stands for the number of trials okay and then p stands for the probability of success in a single trial okay and then one minus p which is the same as q represents the probability of failure in a single trial okay and then x is the number of successes okay so this is the formula we&#39;re using binomial distribution so i&#39;m going to write mine this way and combination x p exponent x then k exponent n minus x please remember q is the same as one minus p and then have written n combination x here still the same as this okay so but i&#39;ll be writing mine in this form so you can choose to write them in any form but i&#39;ll be writing mine like this okay now before i proceed in binomial distribution to find the mean the mean is referred to as expected value of what of x the mean is equal to n times p number of trials times the probability of success in a single triad okay so that will give you the mean and also to find the variance that is variance of x the formula is n times p times q or times 1 minus p so that&#39;s how we find the variance and then to find the standard deviation that is sd of x it should be the square root of the answer you got for the variance so square root of n times p times q that would be the standard deviation so let&#39;s take some questions and see how to do this properly now question one says a point is tossed five times find the probability of getting exactly two tails now to solve this question first what is the binomial formula the formula says that probability of x equals x should be equal to n combination x isn&#39;t it p exponent x times q exponent n minus x now from the question the question says a point is tossed five times it means that the question has given you n to be wrought to be 5 find the probability of getting exactly two tails every question will give you the n and the p and ask you to find the x okay now with the help of the p you can know the kill yourself but yeah like the case that when you look at this question it looks like the p is not in but it is in let&#39;s analyze this you said a coin is paused five times now you ask yourself when i toss a coin once okay when i toss a coin once there are two possible outcomes isn&#39;t it so since there are two possible outcomes the probability of getting a tail in a single tossing should be number of tails in a single torsion you know in a single tossing there are two possible outcomes tail or head so probability of getting a tail in a single torsion should be number of tails in a single thousand divided by number of outcomes in a single torso which is two so we&#39;ve gotten p to be one over two and then together q the q is one minus p isn&#39;t it which is equal to one minus our p was one over two so that is going to give us 1 over 2 also so we can now substitute in the formula but the question says find the probability of getting exactly two tails so to find the probability of getting exactly two that x is exactly two what we say exactly means that it is equal to two okay will give us what is our end according to the question it is five so five combination x is two so combination two then rp is one over two which is the same as zero point five okay so 1 over 2 we can write it 0.5 1 over 2 exponents 2 times our q is also 1 over 2. now n is 5 according to the question isn&#39;t it and then our x is 2. so we&#39;re going to have now when you put 5 combination 2 on your calculator to get it first enter five okay then click on shift and then the division sign you get a combination now i get a combination it comes as a symbol c then you click on two so five combination 2 give you 10 times 1 over 2 exponent 2 will give you 1 over 4 that is 1 squared divided by 2 squared okay you can use your calculator and then five minus two is three isn&#39;t it one over two exponent three should give you one over eight now when you simplify this you get what five divided by what 16. so in decimals it is 0.3125 leave your answer in four decimal places okay so in decimal it is 0.3125 so this is the answer to the question now if the question had asked you that you should find the mean okay the mean which is expected value of x should be equal to what n times p and what is our n our n is five and our p is what one over two so five times one over two should give us what two point five so the mean is what is two point five for the variance that&#39;s variance of x it should be n times b times q isn&#39;t it so our n is what five isn&#39;t it and then p is one over two and q is one over two so five times one over two times one over two should give us what one point two five so this is the variance so the variance is what 1.25 if you want to find a standard deviation then a standard deviation should be what square root of what you got for the variance so square root of 1.25 so square root of 1.25 will give you us one point so this is the mean 2.5 and the variance is 1.25 and the standard deviation is 1.118 the question only asks for the probability of getting exactly 2 too so this is the answer 0.3125 so let&#39;s proceed to the next question question 2 says it is known that 37 of inhabitants of a community favor a political party pa so this is the name of that political party now a random sample of 30 inhabitants was selected from the community and each person was asked if he or she will vote for pa party in an impending election what is the probability that hey no one will vote for pa party so let&#39;s see how to solve this now for this question rp is 37 so you have to convert it towards decimals or fraction okay so 37 divided by 100 will give us 0.37 okay then our q is going to be 1 minus p which is 0.37 so 1 minus 0.37 give us 0.63 okay so that&#39;s our kill and then from the question they said a random sample of 30 so this 30 is the n the sample size okay so that&#39;s now to solve question a question a says what is the probability that no one will vote for party it means that we should find the probability that x will be exactly zero no one will vote for party a it means x will be out exactly zero so to solve this what is our n our n is what 30 so we say 30 combination x which is zero then rp is what rp is 0.37 so we say 0.37 exponent x which is 0 times q is 0.63 so 0.63 exponent 30 minus 0 that is n minus x okay so let&#39;s simplify this you can put this on your calculator directly okay now 30 combination zero will give you one times any number exponent zero also give you one okay times zero point three exponent 30 minus 0 is k80 so if you simplify this with your calculator you&#39;ll get 0.0000955 so that would be the answer okay now let&#39;s look at question b question b says we should find the probability that exactly two persons will vote for ba party so we want to find the probability that x is exactly 2. so that will be what n combination 2 that&#39;s n combination x then p exponent 2 times q which is 0.63 exponents 30 minus 2 now you can put this direction on your calculator to do that you input 30 then you click on shift and the division key and that will give you the combination symbol then you enter 2 then you bring your multiplication sign times open a bracket and then enter 0.37 in the bracket close it and square it then bring times enter 0.63 in the bracket and then exponent 30 minus 2 so just click equal to sign you get answer straight forward so 0.00014 so that&#39;s the answer to your question b okay now question c says you should find a probability that at most two persons will vote for pa pattern so question c says at most two persons now when we say at most at most means that maximum should be that value so i would say at most two it means that maximum should be two it shouldn&#39;t be more than two okay so at most two means that x is less than or equal to two it means that it should either be less than two or equal to two but not more than two that&#39;s the mean of at most two okay so if we say x is less than or equal to 2 then it means that it&#39;s either x is exactly zero or x is exactly one or x is exactly true that is the meaning it&#39;s either that x is exactly zero or x is exactly one or existing do you get it so if i say i go to the studio at most twice in a year it means that it&#39;s either i don&#39;t go at all or i go just once in a year or i go there twice in a year but not more than twice that is the meaning okay so to solve this we are going to use the binomial formula to find the probability that x is exactly zero okay then you use that same formula to find the probability that x is exactly one use that same formula to find the probability that x is exactly 2. so to do that we can say what n combination was x which is 0 then p which is 0.37 0 times 1 minus p which is 0.63 exponents 30 minus 0 then plus what you&#39;ve written here is just for probability that x is exactly zero so you find for this as well 30 combination one zero point three seven exponent one then zero point six three exponent 30 minus 1. now this thing that the probability of success remains the same for this and this again 30 combination 2 0.37 exponents 2 then 0.63 exponent 30 minus 2. so when you put this on your calculator you&#39;re going to get zero zero 0.0000 nine five five plus when you solve the second one also you&#39;re going to get zero point zero zero zero zero two plus the last one is going to be 0.00014 so when you add the three of them together you&#39;re going to get zero 0.0001609 five five so you can approximate it to zero point zero zero zero one six so that is the answer to question c so let&#39;s look at question d now question this is what is the probability that at least at least three persons will vote for pa party now when we say at least three it means from three upwards okay at least many the minimum should be three doesn&#39;t mean so when we say at least three means that x is greater than three or equal to three okay at least three it means that three or more so greater than or equal to three okay now to find at least three is going to be probability that x is exactly three because we say three or more right or probability that x is exactly 4 or probability that x is exactly 5 you do that up to n okay up to probability that x is exactly 30. but it&#39;s not going to be easy to find everything out okay okay so for that matter you need to know another method of doing this now to understand the second method of doing this there&#39;s something i want you to know now just like how you learn words and opposites that the opposite of good is bad the opposite of open is closed and the lies now i want to show you some ways and opposite here and i&#39;ll help you to solve this question in a simple way okay now and it goes like this look at the sign less than or equal to sign each opposite is greater than please don&#39;t include the equal to in the opposite because it&#39;s already included here so the opposite of greater than sign should be less than or equal to okay and also the opposite of greater than or equal to should be less than and also the opposite of less than is greater than or equal to it goes in that manner also the equal to sign its opposite is not equal to okay so if you notice that will help you to get a simple method to solve this question so it goes like this probability that x is greater than or equal to three should be equal to one minus its opposite so what is the opposite of greater than or equal to it is less than isn&#39;t it so it should be one minus probability that x is less than three if you&#39;re able to write this you are gone so this will give us one minus so let&#39;s break this down x less than three means from zero to two okay less than three not less than or equal to three but less than three okay so it means you open a bracket probability that x is exactly zero plus probability that x is exactly one plus probability that x is exactly two so that will give us one minus into bracket now from the answer to question c we got a probability that x is actually zero to be 0.000 zero zero zero zero nine five five so there&#39;s no need to solve anything you can just take it from there okay plus probability that x is only one is zero point zero zero zero zero two plus probability that x is exactly 2 is 0.00014 so this will give us 1 minus when you add them together you get 0.0001 so 1 minus 0.00016 will give you what 0.99984 so this is the answer to question d okay 0.99984 now in case your question asks you to find the mean variance and standard deviation to get the mean the mean which is expected value of x will be equal to n times p square n is 13 and our p is 0.37 so 30 times 0.37 will give us 11.1 okay and then the variance will be n times p which is 0.37 times q which is 0.63 so when you multiply it you get 6.993 and then for the standard deviation the standard deviation will be the square root of the variance which is 6.993 okay so you get 2.644 so that&#39;s the answer to this so let&#39;s look at another question question 3 says a multiple choice test consists of 15 questions so for this question it means that n is 15 each with 5 possible answers of which only one is correct so it means that the probability of getting one correct should be one over five because you have five possible answers and only one is correct so the probability of getting one correct should be what one over five okay if it contains four possible answers then probability of getting one correct should be worth one over four but this has five possible answers so the probability of getting one correct should what one overhead five okay so we know our n and we know our p now let&#39;s proceed suppose one of the students taking a test answers the questions by guessing what is the probability that he answers at most three questions correctly now when we say at most at most means the maximum should work three isn&#39;t it so we want to find the probability that x is less than or equal to three so that will give us probability that x is exactly zero or x is exactly one or x is exactly two or x is exactly three because they say at most three mini maximum should be three okay then let&#39;s find the probability that x is exactly zero now our n is 15 isn&#39;t it so that should be 15 combination zero and then probability of success is one over five so one over five exponent zero and then probability of failure should be cubed which is one minus one over five which will give us four over five so probability of failure is four over five so four over five exponent 15 minus zero then plus then we do that for probability that x is exactly one as well so we&#39;re going to have 15 combination one one over five exponent one times four over five exponent fifteen minus one plus fifteen combination two times one over five exponent two times 4 over 5 exponent 15 minus 2 plus 15 combination 3 1 over 5 exponent 3 times what 4 over 5 exponent 15 minus 3 we are following the binomial formula okay so when you put this first one on your calculator you&#39;re going to get 0.03 five one eight plus when you put those also on your calculator you get zero point one three one nine four plus when you put the source on your calculator you get zero 0.2309 plus when you put the last one on your calculator you get 0.25014 so when you add the four of them together you&#39;re going to get zero 0.64816 so this is the answer to question 3. in case the question asks you to find the mean variance and standard deviation to get a mean that&#39;s expected value of x the mean should be number of files which is 15 times probability of success which is one over five so that will give you three so the mean is three okay if you want to find the variance variance should be what number of channels times probability of success times probability of failure which is four over five so when you simplify these two you get two point four then to get a standard deviation the standard deviation will be square root of the variance so square root of two point four will give you one point five four nine two okay the answer to this question at most three questions okay is 0.64816 so let&#39;s look at another question question 4 6 a report from the secretary of health and human services stated that 70 percent of single vehicle traffic fatalities that are occurred at night on weekends involve an intoxicated driver if a sample of hitting single vehicle traffic fatalities our care at night on a weekend is selected find the probability that exactly 12 involve a driver who is intoxicated now to solve this the probability of success okay is 70 when you take 70 divided by 100 you get 0.7 or 7 over 10 okay then probability of failure which is killed should be 1 minus 0.7 which gives you 0.3 and then the number of trials they say if a sample of 15 so the sample of 15 is the end okay so our n is what 15 so to solve this probability that exactly 12 involve a driver who is intoxicated so the subject matter is intoxicated that&#39;s what i&#39;ll be was n combination x so our n is what 15 combination x which is 12. okay remember n combination x can also be written as 15 and write it this way combination 12 okay or you can write it this way also in form of vector then p exponent x so our p is 0.7 exponent x which is 12 times r q is 0.3 isn&#39;t it exponent n minus x now when i take 15 combination 12 you&#39;ll get 455 times now 0.7 exponent 12 should give you approximately 0.01384 times now fifteen minus twelve is three isn&#39;t it zero point three exponentially should give you zero point zero two seven so when you multiply the three of them you get 0.17004 so that is the answer to this question as well now it&#39;s because i&#39;ve approximated some of the values i&#39;m getting this if you had put this part on your calculator straight forward maybe you&#39;d have gotten something slightly different from the it could be 0.17002 okay that doesn&#39;t make your answer wrong okay the answer is still correct okay so in case the question asks you to find the mean you just take the number of charge times probability of success that is 15 times 0.7 you get 10.5 okay if you want to find a variance then it should be 15 times b which is 0.7 times q which is 0.3 and that will give you 3.15 to find the standard deviation you take the square root of what you got for the variance okay so square root of 3.15 will give you 1.775 so that is the answer so let&#39;s take another question question 5 says a die is rolled 480 times so solution our n is 480 so meaning number of trials is 480. find the mean variance and standard deviation of the number of trees that will be ruled now when you look at this question you do not have p isn&#39;t it but for this particular question you can get a p yourself to get a p you ask yourself when a roller die once what is the probability of getting a 3 in a single rolling okay so the probability of getting a three in a single ruling should be one over six the reason is there is only one three in a single ruling isn&#39;t it and there are six possible outcomes in a single rolling of a diet so number of threes divided by what number of possible outcomes which is one over six okay so p is one over six so to answer the question the mean which is expected value of x should be equal to n times p which is 480 times one over six four hundred and eighty times one over six will give you eighty okay then to find the variance this is how write a variance this symbol is called sigma okay it&#39;s a greek letter it&#39;s called sigma it&#39;s a small letter sigma this is a capital letter sigma and this is a small letter one okay so sigma square is a symbol for representative variance so variance of x should be equal to n times p times one minus p or n times p times q okay so that will give us 480 times one over six times what is our q our q is 1 minus 1 over 6 which is 5 over 6 okay so [Music] 5 over 6 now 480 times 1 times 5 will give you 2 400 divided by this is over 1 1 times 6 times 6 will give you 36 you can use your calculator okay when you simplify this you get 200 over three okay if you are leaving enzyme mixed fraction then it should be 66 whole number two over three if you are leaving answer in decimal then it should be 66.667 now to find the standard deviation now the standard deviation the symbol is sigma without a square okay so ld of x should be equal to square root of what you got for the variance okay square root of n times p times q so square root of 66.667 will give you 8.165 so that&#39;s the final answer to this so let&#39;s proceed to the next question question 6 says a large retailer purchases a certain kind of product from a manufacturer the manufacturer indicates that the defective rate of your product is three percent in a shipment the inspector of the retailer randomly pick 20 items of the product from a shipment what is the probability that there will a be three defective items now from this question the probability of success is three percent that&#39;s three over hundred and i&#39;ll give a zero point zero three okay and the number of trials is and then probability of failure which is q should be 1 minus success which is 0.03 so 1 minus 0.03 will give us 0.97 now to solve question a it says if you find the probability that there will be three defective items so we are finding probability that x is exactly three so that will give us n combination x so our n is what 20 so 20 combination 3 times rp is 0.03 so 0.03 exponent x which is 3 times our q is 0.97 exponent n minus x and our n is 20 minus our x is 3 isn&#39;t it so 20 combination 3 if you put this on your calculator you get 1140 times 0.03 exponentially should give you 0.000027 times now 20 minus 3 is 17 0.97 exponents 17 will give you 0.5958 so when you multiply the three of them you get zero points 0.01834 so that&#39;s the answer to question a now question b says what is the probability that there will not be more than two defective items meaning there won&#39;t be more than two so when we say there won&#39;t be more than two it means that the maximum should be two but not more than two okay so in other words this means at most two okay not more than two means at most two okay so probability that x is less than or equal to two should be probability that x is exactly zero or x is exactly one or x is exactly two so let&#39;s find the probability that x is exactly zero that would be n which is 20 combination x which is zero times zero point zero three exponent 0 times 0.97 exponent 20 minus 0 plus then we do that for the second one also 20 combination 1 p is 0.03 exponent 1 which is x times 0.97 exponent 20 minus 1 then plus 20 combination 2 times 0.03 exponential times 0.97 exponent 20 minus 2. so let&#39;s simplify this you can input this on your calculator straight forward okay now when you solve the first one you&#39;re going to get 0.54379 plus when you solve the second one also you get 0.33637 plus then the table will give you 0.09 eight instead okay so when you add the three of them together you&#39;re going to get zero points nine seven eight nine nine so that&#39;s the answer to question b now in case the question asks you to find the mean variance and standard deviation for the mean the mean is going to be n times p that is 20 times 0.03 and that will give you zero 0.6 okay for the variance it should be n which is 20 times p which is 0.03 times q which is 0.97 and that will give you 0.582 and then to find the standard deviation the standard deviation should be square root of the variance and square root of 0.582 should give you 0.763 so let&#39;s proceed to the next question question 76 it is known that all items produced by a certain machine will be defective with probability of 0.1 independently of each other what is the probability that in a sample of three items at most one will be defective now from the question rp is 0.1 our q is going to be 1 minus p which is 1 minus 0.1 and that will give us 0.9 okay and then n is 3. and the question says we should find the probability at most one will be defective so we say at most one meaning the maximum should be one so it means that x is less than or equal to one okay and that will give us probability that x is exactly zero or x is exactly one so let&#39;s find the probability that x is exactly zero and probability that x is a valley one then we add them together so that will give us three combination zero times probability of success is zero point one isn&#39;t it exponent zero times now probability of failure is 0.9 exponent 3 minus 0 plus then we take the second one also three combination one rp is 0.1 isn&#39;t it so 0.1 exponent 1 and an rq is 0.9 isn&#39;t it so 0.9 exponent 3 minus 1. so 3 combination 0 will give you 1 times 0.1 exponents there is also one times zero point nine exponents three minus zero is three okay plus now three combination one will give you three times zero point one exponent will give you 0.1 okay times 0.9 exponent 3 minus 1 is 2 okay now when you simplify 1 times 1 times 0.9 exponent 3 you get 0.729 okay plus 3 times 0.1 times 0.9 exponent 2 will give us 0.243 so when you add the two of them together 0.729 plus 0.243 you get 0.972 so the correct answer for question 7 is 0.972 okay now in case the question asks you to find the mean expected value of x it should be number of child which is three times the probability of success which is 0.1 so 3 times 0.1 gives you 0.3 so that&#39;s the mean and then the variance the variance should be number of childs times probability of success times probability of failure so three times zero point one times zero point slice div zero point two seven and then the standard deviation should be the square root of the variance so square root of zero point two seven is zero 0.52 so let&#39;s proceed to the next question question 8 is suppose the probability that a colored freshman will graduate is 0.6 3 college freshmen are randomly selected what is the probability that at most two of these students will graduate so to solve this probability that one person will graduate is 0.6 isn&#39;t it and the probability that one person will not graduate which skill should be 1 minus 0.6 which is 0.4 but the question says three college freshmen are randomly selected so n is what is three and also the question is saying that we should find the probability that at most two of these students will graduate so what we say at most so it means that x is less than or equal to two which will give us probability that x is exactly zero or x is exactly one or x is exactly two okay so let&#39;s find a probability that x is exactly zero and that should be what n combination x which is three combination zero then p exponent x times q exponent n minus x plus n combination 1 this time around times p which is 0.6 exponent 1 times k which is 0.4 exponent 3 minus 1 plus n combination 2 p exponents 2 times 0.4 exponent 3 minus 2. so you can put all this on your calculator you get 0.064 okay plus so when i simplify this also you get 0.288 [Music] plus then when you simplify the third one you get 0.432 so add the three results together when you add them you get 0.784 so the correct answer to question 8 is 0.784 in case the question asks you to find the mean variance and standard deviation the mean is going to be n which is 3 times p which is 0.6 so 3 times 0.6 will give you 1.8 okay to find the variance the variance should be 3 times 0.6 times the probability of failure which is 0.4 so 3 times 0.6 times 0.4 will give you 0.72 okay and then to find the standard deviation it should be the square root of the variance so square root of 0.72 gives 0.849 now this is another style of giving you a binomial question now look at this the question says the random variable x has binomial distribution b into bracket 40 comma 0.2 determine each of the following a probability that is exactly five going okay now the nature of this question is different from the previous ones that i have solved okay but what i want you to know is that whenever you have this type of question what you should know is that the first argument here is the number of childs which is 40 and then the second argument is 0.2 should be the probability of success so now you know that n and then you know the p so you can easily find the q as what one minus 0.2 which will give you 0.8 so with this known and we p also known and q known you can solve a b c d e and f okay so i want you to try this so i&#39;m going to give you the answers so that when you try then you are correct you know now for question a i think you get 0.08541 and then for question b you get 0.95676 then for question c you&#39;re going to get 0.1607 and then for question d you&#39;re going to get 0.24881 and then question e you get 0.02846 and then for question f i won&#39;t tell you the answer try as much as possible to get an answer but make sure you work with a to e properly to check if your answer is the same as mine so i&#39;ll end the binomial distribution here for now in our next video we are going to look at the poisson distribution so if you haven&#39;t subscribed to the channel yet please do also subscribe so that whenever i post more important videos like this you can always have access to it easily thank you [Music] [Applause] [Music] [Applause]

Transcript for:Understanding Binomial Distribution Concepts

Transcript for:
Understanding Binomial Distribution Concepts