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# Compound probability of independent events

AP.STATS:
VAR‑4 (EU)
,
VAR‑4.E.1 (EK)

## Video transcript

let's think about the situation where we have a completely fair coin here so let me draw it I'll assume it's a quarter or something let's see so this is a quarter let me draw my best attempt at a profile of George Washington well that'll do so it's a fair coin and we're going to flip it a bunch of times and figure out the different probabilities so let's start with a straightforward one let's just flip it once so with one flip of the coin what's the probability of getting heads well there's two equally likely possibilities and the one with heads is one of those two equally likely possibilities so there's a 1/2 chance same thing if we were to ask what is the probability of getting tails there are two equally likely possibilities and one of those gives us tails so 1/2 and so and this isn't one thing to realize if you take the probabilities of heads plus the probability of tails you get 1/2 plus 1/2 which is 1 and this is generally 2 the probability the sum of the probabilities of all of the possible events should be equal to 1 and that makes sense because you're taking you're adding up all of these fractions and the numerator will then add up to all of the possible events the denominator is always all of the possible events so you have all the possible events over all the possible events when you add all of these things up now let's take it up a notch let's figure out the probability of I'm going to take this coin and I'm going to flip it to twice the probability of getting a heads and then getting another heads the probability of getting a head and then another head so there's two there's two ways to think about it one ways to just think about all the different possibilities I could get a head on the first flip and a head on the second flip head on the first flip tail on the second flip I could get tails on the first flip heads on the second flip or I could get tails on both flips so there's four distinct equally likely possibilities four distinct equally likely outcomes here and one way to think about it is on the first flip I have two possibilities on the second flip I have another two possibilities I could have heads or tails heads or tails and so I have four possibilities for each of these possibilities for each of these two I have two possibilities here so either way I have four equally likely possibility and how many of those meet our constraints well we have it right over here this one right over here having two heads meets our constraint so this is and there's only one of those possibilities I've only circled one of the four scenarios so there's a 1/4 chance of that happening another way you could think about this and this is because these are independent events and this is a very important idea to understand in probability and we'll also study scenarios that are not independent but these are independent events the the what what happens in the first flip in no way affects what happens in the second flip and this is actually one thing that many people don't realize or something called the gamblers fallacy where someone thinks if I got a bunch of heads in a row then all of a sudden it becomes more likely on the next flip to get a tails that is not the case every flip is an independent event what happened in the past in these flips does not affect the probabilities going forward so the probability of getting a heads on the first flip in no way or the fact that you got a heads on the first flip and no way affects that you got a heads on the second flip so if you can make that assumption you could say that the probability of getting heads and heads or heads and then heads is going to be the same thing as getting probability as the probability of getting heads on the first slip times the probability of getting heads on the second flip and we know the probability of getting heads on the first flip is 1/2 and the probability of getting heads on the second flip is 1/2 and so we have 1/2 times 1/2 which is equal to 1/4 which is exactly we got what we got when we tried out all of the different scenarios all of the equally likely possibilities let's take it up another notch let's figure out the probability and we've kind of been ignoring tails so let's pay some attention to tails the probability of getting tails and then heads and then tails so this exact series of events so I'm not saying to you know in any in any order two tails in a head I'm saying this exact order the first flip is a tails second flip is a heads and then third flip is a tail so once again these are all independent events what the fact that I get tails on the first flip and no way affects the probability of getting a heads on the second flip and that in no way affects the probability of tails on the third flip so because these are independent events we can say this is the same thing as the probability of getting tails on the first flip times the probability of getting heads on the second flip times the probability of getting tails on the third flip and we know these are all independent events so this right over here is 1/2 times 1/2 times 1/2 1/2 times 1/2 is 1/4 1/4 times 1/2 is equal to 1/8 so this is equal to 1/8 and we can verify it let's try it all of the different scenarios again so you could get heads heads heads you can get heads heads tails you could get heads tails heads you could get heads tails tails you can get you can get tails heads heads this is a little tricky sometimes you want to make sure you're being exhaustive in all of the different possibilities here you could get you could get tails tails heads tails you could get tails tails heads or you could get tails tails tails and what we see here is that we got exactly eight equally likely possibilities we have eight equally likely possibilities and the tail heads tails is exactly one of them it is this possibility right over here so it is one of eight equally likely possibilities