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# Generalizing k scores in n attempts

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

## Video transcript

so the last video we studied a circumstance where I had a 70 percent free-throw probability I have a 70 percent chance of making any free-throw which is actually higher than my actual free-throw probability which might be a surprise to you but we said in that circumstance if it's a 70 percent chance of making it well that means that you have a 1 - 70 percent or 30 percent chance of missing and we said if you took six attempts the probability of you getting exactly making two of the baskets exactly two scores and I call them scores instead of making it just because I wanted making and miss it have different letters in that video we said well there's six choose two different ways of making two exactly two out of the six free throws and then the probability of any one of those ways is going to be making it twice which is 0.7 squared tie and missing it four times so 0.3 to the fourth power so this was just one particular situation but we could generalize based on the logic that we had in that video in fact let's do that so if I were to generalize it if I were to say the probability the probability it's the exact same logic of exactly exactly now let's say K let me do this in a color and interesting color so let me do it in this orange brown color k k shots or exactly K scores I'll call making a free throw of score we'll just assume you've got a point for it so exactly 2k scores K scores in in n attempts let's just say in n attempts in N and let me go back to that green color and attempts and a temps is going to be equal to well how many ways can you pick K things out of N or n choose K and choose K and choose K and then actually let's just generalize it even more let's just say that you have your free-throw probability is P so let's say let's say P is so for this situation right over here since we generalize it fully let's say that P is the probability of making a free-throw actually since I already have a P here let me just say F is equal to the probability of making making a free throw or you could say your probability of scoring if you call the score making making a free throw so if F is your probability of making a free throw so if you want n scores then this is going to be this is going to be well it's going to be F to the N power and then you're going to have and then you're going to miss the remainder sorry F to the K power because you're making exactly K scores so F to the K power and then the remainder so the N minus K attempts you're going to miss it so it's going to be that probability of missing and the probability of missing is going to be one minus F so it's going to be x times 1 minus F to the N minus K power to the N minus K power and just if you like or I encourage you pause the video and just make sure you understand the parallels between this this example where I had a set where you know I guess our F was 70% our F was 70% 1 minus F or our F was 0.7 and 1 minus F would be 0.3 and we were seeing how do we get two scores in six attempts and here we're saying K scores in n attempts and this is just a general way to think about it and the whole reason why I'm setting this up this way is it's interesting to now think about the probability distribution for a random variable that's defined by the number of scores in your n attempts or the number of scores in your six attempts and actually since I've been pushing the limit in or you know I've been doing longer videos that I attend to I will do that in the next video