Interpreting general multiplication rule

- [Instructor] We're told that two contestants are finalists in a cooking competition. For the final round, each of them spin a wheel to determine what star ingredient must be in their dish. I guess the primary ingredient, and we can see it could be chard, spinach, romaine lettuce, I'm guessing, cabbage, arugula, or kale. And so then they give us these different types of events, or at least the symbols for these different types of events, and then give us their meaning. So K-sub 1 means, the first contestant lands on kale, K-sub 2 means, the second contestant lands on kale, K-sub 1 with this superscript C, which you could view as complement. So K-sub 1 one complement, the first contestant does not land on kale. So it's the complement of this one right over here. And then K-sub 2 complement, would be that the second contestant does not land on kale. So the not of K-sub 2 right over here. Using the general multiplication rule, express symbolically the probability that neither contestant lands on kale. So pause this video and see if you can have a go at this. All right, so the general multiplication rule is just saying this notion that the probability of two events, A and B, is going to be equal to the probability of, let's say A given B, times the probability of B. Now, if they're independent events, if the probability of A occurring does not depend in any way on whether B occurred or not, then this would simplify to this probability of A given B, would just become the probability of A. And so if you have two independent events, you would just multiply their probability. So that's just all they're talking about, the general multiplication rule. But let me express what they're actually asking us to express. The probability that neither contestant lands on kale. So that means that this is going to happen, the first contestant does not land on kale, and this is going to happen, the second contestant does not land on kale. So I could write it this way. The probability that K-sub 1 complement and K-sub 2 complement, and I could write it this way. This is going to be equal to, we know that these are independent events because if the first contestant gets kale or whatever they get it, it doesn't get taken out of the running for the second contestant. The second contestant still has an equal probability of getting or not getting kale, regardless of what happened for the first contestant. So that means we're just in the situation where we multiply these probabilities. So that's gonna be the probability of K-sub 1 complement, times the probability of K-sub 2 complement. All right, now let's do part two. Interpret what each part of this probability statement represents. So I encourage you like always, pause this video and try to figure that out. All right, so first let's think about what is going on here. So this is saying, the probability that this is K-sub 1 complement. So the first contestant does not land on kale. So first, first contestant does not get kale, and, I'll write and in caps, and second contestant does get kale. And second does get kale. So that's what this left-hand is saying. And now they say that that is going to be equal to, so this part right over here, probability that the first contestant does not get kale. Probability that first does not get kale, times, right over here. And the second part right over here is the probability that the second contestant gets kale, given that the first contestant does not get kale. So probability that the second gets kale, given, that's what this vertical line right over here means, it means given, shorthand for given. Given, I wrote it up there too. Given that first does not get kale. And we're done, we've just explained what is going on here.