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In this tutorial, you're going to learn about the general "and" probability formula for dependent events. In its separate tutorial, we talked about "and" probability for independent events, and that had a special multiplication rule. This has a general multiplication rule.
So, let's take a look. Suppose there's a famous television show "Go For Broke", and a person gets to roll a die to determine which jar they're going to draw a colored chip from. And if the chip is green, they win a prize. If the chip is red, they don't win anything. So, if a person rolls an even, they can select from the first jar which contains seven green and 13 red. So, that's not too bad probability to win the famous, fabulous prize.
How about the second one? If a person rolls odd, they select from jar B, which contains five green and 25 red. So, that's not a very good likelihood. You'd rather select from jar A. Because the probability of picking green or red-- suppose we're saying red-- the probability of picking a red chip depends on the die roll. That means that these are dependent. The probability of red increases if you roll an odd number versus if you roll an even number.
Let's look at it in a tree diagram. There's a 1/2 probability of rolling an odd and a 1/2 probability of rolling an even. But that's where the similarities end. Once you go there, there's a five out of 30 probability that you get the green chip, if you roll the odd, and a 25 out of 30 probability of getting the red chip. If you roll an even, there's a seven out of 20 probability of getting the green, and a 13 out of 20 probability of getting the red.
In previous examples, we multiplied down the tree branches to find the joint probability of odd and green. But take a look. This number, here, is the probability of selecting a green if you've already rolled an odd. This is a conditional probability. This is the probability of green given odd.
So, now what are we multiplying? We're multiplying the probability of an odd times the probability of green, given odd. And this leads us to the general probability formula, the "and" probability formula for dependent events. It's the probability of the first event-- in this case, O-- times the probability that the second event, green chip, occurs given that we're already down that road on the tree diagram. When we multiply it all out, we end up with 1/12 here, 5/12 here, 7/40 here, and 13/40 here. You multiply the probability times the probability of green given odd, or the probability of red given odd. Or the probability of green given even, or red given even.
So, let's look again at what independence really means. We said that knowing the outcome of the first event, A, knowing whether or not it happened, doesn't change the probability of the second event. So, using symbolic conditional probability notation, we can say the probability that B occurs given that we know that A happened is the same as just the probability of B, whether or not we knew A was happening. So, let's take a look. The probability of B knowing that occurred is the same as the probability of B if we don't know whether or not A has occurred. So, A having no effect on the probability of B.
And so what does that mean within the formula? That means the probability of A and B is equal to this by the general multiplication rule. But the probability of B given A is the same as the probability of B. This formula looks familiar. That's the special multiplication rule for independent events. So, it makes a lot of sense. The special multiplication rule is a special version of the general multiplication rule.
So, let's recap. The joint probability of two events occurring together, either concurrently or consecutively, is equal to the probability of the first event times the probability of the second event given that the first event occurred. It's best to think about this like you're going down the branches of a tree diagram. Given that I've already got where I've gotten, what's the probability of going down this next branch? And the general formula works for both independent and dependent events. So, we talked about the general multiplication rule, which is a lot more powerful than the special multiplication rule. Good luck and we'll see you next time.
The probability that two events both occur is the probability of the first event times the conditional probability that the second event occurs, given that the first already has.