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now that we've spent a couple of videos exploring a scenario where I'm taking multiple free-throws and figuring out the probability of making K of the scores in six attempts or in n attempts let's actually define a random variable using this scenario and and see if we can construct its probability distribution and we'll actually see that it's a binomial distribution so let's define the random variable X so let's say that X is equal to the number the number of made shots number of made free throws when taking when taking six free throws so it's how many of the six do you make and we're going to assume what we assumed in the first video in this series of this these making free throws so we're going to assume the 70 percent free throw probability right over here so assuming assumptions assuming 70 percent free-throw free-throw percentage all right so let's figure out the probabilities of the different values that X could actually take on so let's see what is the probability what is the probability that X is equal to zero that even though you have a 70 percent free-throw free-throw percentage that you make none of the shots and actually you could you could calculate this through to probably some common sense without using all of these fancy things but just to make things consistent I'm going to write it out so this is going to be this is going to be it's going to be equal to six choose zero times zero point seven to the zeroth power times zero point three to the sixth power and this right over here is going to end up being one this over here is going to be end up being one and so you're just going to be left with zero point three to the sixth power and I calculated it ahead of time so if we just round to the nearest if we do if we were on our percentages to the nearest tenth this is going to give you approximately approximately well the decimal to the nearest to the nearest thousandth you're going to get something like that which is approximately equal to zero point one percent chance of you missing all of them so you know roughly MSB roughly here one in a thousand one in a thousand chance of that happening of missing all six free throws now let's keep going this is fun so what is the probability that our random variable is equal to one well this is going to be six choose one times zero point seven to the first power times zero point three to the to the six minus first power so it's a fifth power and I calculated this and this is approximately 0.01 or we could say one percent so still a fairly low probability ten times more likely than this roughly but still a a fairly low probability let's keep going so the probability that X is equal to - well that's what our first video was essentially so this is going to be six choose two times zero point seven squared times zero zero point three to the fourth power and we saw that this is approximately going to be zero zero point zero six or we could say six percent and obviously you know you could type these things in a calculator and get a much more precise answer but just for the sake of of just getting a sense of what these probabilities are look like and that's why I'm giving these these rough estimates kind of I guess you could say to the close to the closest maybe tenth of a tenth of a percent and actually even if you round to the closest tenth of a percent you actually get two 6.0% and this is one point one point Oh percent because this we actually went to a tenth of a percent here but let's keep going we're obviously going to have to do a few more of these so let me just make sure have enough real estate all right so the probability that our random variable is equal to three is going to be six to three and I'm sure you could probably fill this out on your own but I'm going to do it 0.72 the third power times 0.3 to the 6 minus 3 which is a third power which is approximately equal to well it's going to be zero point one eight five or eighteen point five eighteen point five percent so yeah you know that's definitely within the realm of possibility I mean all of these are in the realm of possibility but it's starting to be a non insignificant probability now let's do the probability that our random variable is equal to four well it's going to be six choose four times zero point seven to the fourth power times zero point three to the 6 minus four or second power which is equal to this is going to get equal to or approximately because I am taking away a little bit of the precision when I write these things down zero point three to four so approximately 32 point four percent chance of making exactly four out of the six free throws all right two more to go let's see I have not used purple as yet so the probability that our random variable is equal to five it's going to be six choose five or zero at times I should say zero point seven to the fifth power times zero point three to the first power and that is going to be roughly roughly zero point three zero three which is 30.3% that's interesting one more left so the probability that I make all of them of all six is going to be equal to is equal to six choose six and zero point seven to the six power times the zero point three to the zeroth power which is this right over here is going to be one this is going to be one so it's really just zero point seven to the sixth to the sixth power and this is approximately zero point 1 1 8 I calculated that ahead of time which is eleven point eight eleven point eight percent and so there's something interesting that's going on here the first time we looked at the binomial distribution we said hey you know there's the symmetry as we kind of got to some type of a peak and went down but why don't see that symmetry here and the reason why you're not seeing that symmetry is that you are more likely to make a free-throw than not is so you have a seventy percent free-throw probability here this is no longer just flipping a fair coin where you will see the symmetry is in these coefficients if you calculate these coefficients six choose zero is one six choose six is one you would see that six choose one is six and six choose five is six you'd see six choose two is fifteen and six twos 4 is also 15 and then 6 choose 3 is 20 so you definitely see you definitely see the symmetry in the coefficients but then these things are weighted by the fact that you're more likely to make something than miss something if if these were both point five then you would also see the symmetry right over here and you can plot this to essentially visualize what the probability distribution looks like for this example and I encourage you to do that to take these different cases just like we did in that first example with the fair coin and plot these but this essentially does give you the probability distribution for for the random variable in question this is I just wrote it out instead of just visualizing it but it says okay well well so these are these are the different values that this random variable can take on it can't it can't take on negative one or it can't be 15.5 or PI or or 1 million these are the only seven values that this random variable can take on and I've just given you the probabilities or I guess you could say the rough probabilities of the random variable taking on each of those each of those seven values

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