Independent versus dependent events and the multiplication rule
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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 this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1/2 plus 1/2, which is 1. And this is generally true. The sum of the probabilities of all of the possible events should be equal to 1. And that makes sense, because 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 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 twice-- the probability of getting heads and then getting another heads. There's two ways to think about it. One way is to just think about all of 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. And one way to think about 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 possibilities. 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 constraints. 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. 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. There's something called the gambler's 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. The fact you got a heads on the first flip in 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 the probability of getting heads on the first flip 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 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 in any order two tails and 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. The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we could say that's 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 out all of the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. 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 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 1 of 8 equally likely possibilities.
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