How do you calculate P(A/B), when event B occurred after event A?

Question

There's really only one question I can't begin to handle when it comes to probability, literally. It's not the only type of question I struggle with, though it's the type of question where I can't even get a foothold.

Here's an example:

On a randomly selected day, the probability that Jacques will eat cereal for breakfast, P(C), is 0.4, the probability that he will eat a sandwich for lunch, P(S), is equal to 0.6, and the conditional probability that he will eat a sandwich for lunch, given that he has already eaten cereal for breakfast, P(S | C), is equal to 0.7.

Based on this information, what is P(C | S), the probability that Jacques eats cereal for breakfast given that he eats a sandwich for lunch?

Soruce: Khan academy.

Ok, so first of all, I know that P(S | C) is 0.7, because regardless of P(C), P(S) given P(C) happened anyway, is 0.7. However, I cannot for the life of me figure out a calculation for the probability that he eats cereal for breakfast given that later on he's going to eat a sandwich.

The only thing I can think to include is the average Probability that he eats a sandwhich, which is 0.6, but apart from that...

Any help would be much appreciated.

By the way, do you think this is a little advanced for a beginner in stats? To me it's like tackling multiplication when your in the middle of learning addition.

Answer 1

Bayes' rule

$\mathbb P(C\mid S)$ = $\frac{\mathbb P(S \mid C) \ \mathbb P(C)}{\mathbb P(S)}$

$\mathbb P(C\mid S) = \frac{(0.7)(0.4)}{x}$

Plug in the value of $\mathbb P(S)$ , and you're done.

Answer 2

Here is a proof that doesn't directly reference Bayes' Theorem. By definition you know that \begin{align*} P(C\mid S) &= \frac{P(S\cap C)}{P(S)} \\ P(S\mid C) &= \frac{P(C\cap S)}{P(C)} \\ \end{align*} Of course $P(S\cap C) = P(C \cap S)$ because these are the same sets. If you clear the denominators in each, you get $$ P(S \cap C) = P(C\mid S)P(S) = P(S\mid C)P(C) $$ The equation on the right has four quantities, three of which are given and one of which is desired. Solve.