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# Expected value of a binomial variable

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

## Video transcript

so I've got a binomial variable X and I'm gonna describe it in very general terms it is the number of successes after n trials after n trials where the probability of success success for each trial is P and this is a safe this is a reasonable way to describe really any random any binomial variable we're assuming that each of these trials are independent the probability stays constant we have a finite number of trials right over here each trial results in either a very clear success or failure so what we're going to focus on in this video is well what would be the expected value of this binomial variable what would the expected value expected value of X be equal to and I will just cut to the chase and tell you the answer and then later in this video we'll prove it to ourselves a little bit more mathematically the expected value of X it turns out is just going to be equal to the number of trials times the probability of success for each of those trials and so if you wanted to make that a little bit more concrete imagine if a trial is free-throw taking a shot from the free-throw line success success is made shot so you actually make the shot the ball went in the basket your probability is let me do this yellow color your probability this would be your free-throw percentage so let's say it's 30% or 0.3 and let's say for the sake of argument that we're taking ten free-throws so n is equal to ten so this is making it all a lot more concrete so in this particular scenario your expected value your expected value if X is the number of made free throws after taking ten free throws with a free throw percentage of 30 percent well based on what I just told you would be N times B it would be the number of trials times the probability of success in any one of those trials times zero point three which is just going to be of course equal to three now does that make intuitive sense well if you're taking ten shots with a thirty percent free-throw percentage it actually does feel natural that I would expect to make three shots now with that out of the way let's make ourselves feel good about this mathematically and we're gonna leverage some of our expected value properties in particular we're gonna leverage the fact that if I have the expected value of the sum of two independent random variables let's say X plus y it's going to be equal to the expected value of X plus the expected value of y that we talked about in other videos and so assuming this right over here let's construct a new random variable let's call our random variable Y and we know the following things about Y the probability that y is equal to one is equal to P and the probability that Y is equal to zero is equal to one minus P and these are the only two outcomes for this random variable and so you might be you might be seeing where this is going you could view this random variable it's really representing one trial it becomes one in a success zero when you don't have a success and so you could view our rent our original random variable X right over here as being equal to y plus y and well you're gonna have ten of these so we're gonna have ten Y's in the concrete sense you could view the random variable Y as equaling one if you make a free throw and equaling zero if you don't make a free throw it's really just representing one of those trials and you can view X as the sum of n of those trials and it actually let me be very clear here when immediately went to the concrete but I really should be saying n wise because I want to stay general right over here so there are n n wise right over here this was just a particular example but I am going to try to state general for the rest of the video because now we are really trying to prove this result right over here so let's just take the expected value of both sides so what is it going to be so we get the expected value of x of x is equal to what's the expected value of all of this thing but by that property right over here this is going to be the expected value of y plus the expected value of y plus and we're going to do this n times plus the expected value of y and we're going to have n of these so we have n and so you could rewrite this as being equal to so this is our n right over here this is n times the expected value of y now what is the expected value of Y well this is pretty straightforward we can actually just do it directly the expected value of Y let me just write it over here the expected value of y it's just the probability weighted outcomes and since there's only two discrete outcomes here it's pretty easy to calculate we have a probability of P of getting a 1 so it's P times 1 plus we have a probability of 1 minus P of getting a 0 well what does this simplify to well 0 times anything that's 0 and then you have 1 times P this is just equal to P so expected value of Y is just equal to P and so there you have it we get the expected value of x is 10 times the expected value or the expected value of x is n times the expected value of y and the expected value of y is P so the expected value of x is equal to n P hope you feel good about that
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