Interpreting general multiplication rule | Probability & combinatorics

We're told that two contestants are finalists in a cooking competition. For the final round, each of them spins a wheel to determine what star ingredient must be in their dish. I guess the primary ingredient could be charred spinach, romaine lettuce, cabbage, arugula, or kale.

Then, they give us these different types of events, or at least the symbols for these different types of events, and they 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, means the first contestant does not land on kale. So, it’s the complement of this one right over here. 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 the probability of A given B, which would just become the probability of A.

So, if you have two independent events, you would just multiply their probabilities. That’s just all they’re talking about with the general multiplication rule. But let me express what they are actually asking us to do: express the probability that neither contestant lands on kale.

This means that the first contestant does not land on kale, and the second contestant does not land on kale. I could write it this way: the probability that k sub 1 complement and k sub 2 complement.

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 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 a situation where we multiply these probabilities. That’s going to be the probability of k sub 1 complement times the probability of k sub 2 complement.

All right, now let’s do part 2. 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. This is saying the probability that this is k sub 1 complement, so the first contestant does not land on kale. So, the first contestant does not get kale.

And, in caps, the second contestant does get kale. So that’s what this left hand is saying. Now, they say that that is going to be equal to, so this part right over here: the probability that the first contestant does not get kale, probability that the first does not get kale times...

Right over here, the second part is the probability that the second contestant gets kale given that the first contestant does not get kale. So, the probability that the second gets kale given that—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 the first does not get kale. And we’re done—we’ve just explained what is going on here.