عمومی و قانون

Generalized AND Rule

So now we are in a position where we can talk about the generalized And rule. So first of all, if events A and B are independent, you’ll recall then we can use the simple And rule. That just means we can multiply the probability of A times the probability of B. What happens if A and B are not independent?

So first, let’s, let’s figure out exactly what this would mean. So recall if A and B are independent, that means that one doesn’t influence or affect the other at all. So the probability of A happening has absolutely nothing to do with whether B happens or not and vice versa. So one happening does not change the probability of the other, that is independent.

Well what would the opposite of that be? If events A and B are not independent, then whether one happens changes the probability of whether the other happens. So if we know A does happen, then B has a different probability than it would have if A didn’t happen. That would be not independent.

We handle such situations with something called conditional probabilities. So here’s the notation. We use this symbol, P, A and then a vertical line, and then B. Well, what does this mean? This means, assuming that, for whatever reason, we know that event B is true, then given this condition, what is the probability that A happens?

Or more briefly, and this is the way that people usually would say it, what is the probability of A, given B? So, B is the condition, B is the condition we impose, and then A is the thing we want to find out the probability of given that we know that B has already happened. So, here’s an example. Suppose A equals the event a randomly chosen person is male, and B is the selection pool is the United States Senate.

Well, if I just pick A by itself, just ignore B, just the general population. I’m picking a randomly selected person, this is gonna be something very close to 50%. I think technically, women might be 51%, men might be 49% in the general population, but it’s something very close to 50%. But, things really change if we can figure out the conditional probability.

So given the condition that the person I’m picking is from the Senate, what is the probability that I randomly pick a male? Well in the current Senate, the current United States Senate, as it happens, there are 80 men and 20 women. And so, the probability of picking a male is .8. Now of course, in a perfectly fair world, in a perfectly just and fair world with no sexism, this also would equal 50%.

The Senate would be split 50, 50, and it would be representative of a gender. That’s not the world we live in, unfortunately. There are many, many more men in the Senate, and this is why the conditional probability is different from just the probability of A. And another way to say that is this is why these two events are not independent because one changes the, the probability of the other.

Once we have the idea of conditional probability, we can state the generalized And rule. So here are the generalized And rules, notice that the word and still means multiply. That’s a really good tip to hold onto, the word and in probability does mean multiply.

What changes is what we’re multiplying. Now we’re no longer multiplying simply probability of A times probability of B. That was a simple And rule. Now when things are not independent, we have to use these more generalized And rules. So I realize these are scary looking formulas.

In the next video, I’ll show how to apply these.