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# Binomial probability example

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.B (LO)
,
UNC‑3.B.1 (EK)

## Video transcript

let's say that you know your probability of making a free-throw you know that the probability the probability and let's say the probability of scoring or free-throw just because I might say make and miss that they both start with em and look at it'll get confusing so let's say the probability of a scoring score you know free-throw is equal to is going to be let's say 70% if we want to write it as a percent or we could write it as 0.7 if we write it as a decimal and let's say the probability of missing a free-throw then and this is just going to come straight out of what we just wrote down so the probability of missing of a missing a free-throw is just going to be 100% minus this you're either going to make or omit your they're going to score or miss I want to use make in Miss because they both start with them so this is going to be a 30% probability or if we write it as a decimal 0.3 1 minus 0.7 these these are the only two possibilities so they have to add up to 100% or they have to add up to 1 now let's say that you are going to take six attempts and what we're curious about what we're curious about is the probability of exactly exactly two scores two scores in six attempts in six in six attempts so let's think about what that is and I encourage you if you get inspired at any point in this video you should pause it and you should try to work through what we're asking right now so this is what we this is what we want to figure out the probability of exactly two scores in six attempts so let's think about the way let's think about the particular ways of getting two scores in six attempts and think about the probability for any one of those particular ways and then we can think about how well how many ways can we get two scores in six attempts so for example you could get you could make the first two free-throws so it could be score score and then you miss the next four so score score and then it's miss miss miss and miss so what's the probability of this exact thing happening this exact thing well you have a 0.7 chance of making this off scoring the on the first one then you have a 0.7 chance of scoring on the second one and then you have a 0.3 chance of missing the next four so the probability of this exact circumstance is going to be what I'm writing down and hopefully don't get the multiplication symbols confused with the decimals I'm trying to write them a little bit higher so times 0.3 and what is this going to be equal to well this is going to be equal to this is going to be zero point seven squared times zero point three times zero point three to the one two three fourth to the fourth power now is this the only way to get two scores in six attempts no there's many ways of getting two scores in six attempts for example maybe you miss the first one the first attempt and then you make the second attempt you score then you miss the third attempt and then you let's say you make the fourth attempt and then you miss the next the next two so then you miss and you miss this is another way to get two scores in six attempts and what's the probability of this happening well as well see it's going to be exactly this it's just from multiplying in different order this is going to be zero point three times times zero point seven you have a 30% chance of missing the first one a 70% chance of making the second one and then times zero point three 30% chance of missing the third time's a 70% chance of making the fourth times a 30% chance for each of the next two misses if you wanted this exact circumstance this exact circumstance this is once again going to be 0.7 if you just rearrange the order which you're multiplying this is going to be 0.7 squared times zero point three to the fourth power so for any one of these particular ways to get exactly two scores in six attempts the probability is going to be this so the probability of getting exactly two scores in six attempts well it's going to be any one of these probabilities times the number of ways you can get six scores times the number of ways you can get two scores in six attempts well how if you if you have out of six attempts you're choosing two of them to have scores how many ways are there well as you can imagine this is a combinatorics problem so you could write this as you could write this and let me see how I could you're going to take six attempts so you could write this as six choose or we're trying to fight a few you're picking from six things your six attempts and you're picking two two of them or two of them are going to be neat or need to be made if we want to make meet these circumstances it's going to tell us the number of different ways you can make two scores in six attempts and of course we can write this in kind of the binomial coefficient notation we could write this as six choose two six choose two and we could just apply the formula for combinations so and if this looks completely unfamiliar I encourage you to look up combinations on Khan Academy and we go into some detail on on the reasoning behind this formula to actually go makes makes a lot of sense this is going to be equal to six factorial over 2 factorial over 2 factorial times 6 minus 2 factorial so 6-2 6-2 factorial I'll do the factorial in green again 6 minus 2 factorial what's this going to be equal to this is going to be equal to 6 times 5 times 4 times 3 times 2 and I'll just store in the 1 there although it doesn't change the value over 2 times 1 over 2 times 1 and 6 minus 2 is 4 so that's going to be 4 fact or ray'll so this right over here is 4 factorial so times 4 times 3 times 2 times 1 well that and that is going to cancel and then let's see 6 divided by 2 is 3 so this is 15 there's 15 different ways that you could get that you could you could pick 2 things out of 6 I guess there's one way to say it or there's six there's there's 50 did I say 60 no there's 15 then the 6 is in the 5 there's 15 different ways that you could pick 2 things out of 6 or another way of thinking about it is there's 15 different ways to make two out of six free throws now the probability for each of those is this right over here so the probability of exactly two scores in six attempts well this is where we deserve a little bit of a drumroll this is going to be this is going to be six choose two times zero point seven zero point seven squared that's this is to two you're going to make two you're going to make two and then it's 0.3 0.3 to the zero point three to the fourth power notice these will necessarily add up to six so this right over here was a three then this right over here would be a three and then this would be 6 minus three or three right over here and now what is this value well it's going to be equal to it's going to be equal to we have our 15 3 times 5 so we have this business right over here it's going to be 15 times times let's see in yellow 0.7 times 0.7 is going to be times 0.49 and let's see 3 to the fourth power would be 81 3 to the fourth power would be 81 but I'm multiplying four decimals each of them have one space to the right of the decimal point so I'm going to have this is going to this is I'm going to have four spaces to the right of the decimal so zero point zero zero eight one so there you go whatever this number is and actually I might as well get a calculator out and calculate it so this is going to be this is going to be me so it's 15 times zero point four nine times zero point zero zero eight one and we get zero point five nine five three five so this is going to be equal to let me write it down and actually maybe I'll well I wish I had a little bit more real estate right over here but I'll write it in a very bold color this is going to be well actually I'm kind of out of bold colors I'll write it in a slightly less bold color this is going to be equal to zero point zero five nine three five if we wanted the exact number or we could say this is approximately if we round to the nearest percentage this is approximately a six percent chance six percent probability of getting of getting exactly two scores in the six attempts I didn't say two or more I just said exactly two scores in the six attempts and it actually it's a fairly low probability because I have a pretty high free-throw percentage if someone has this high of a free-throw percentage it's actually reasonably unlikely that they're only going to make two scores in the six attempts