Question about conditional probability.

Question

This is a probability textbook, and this is about conditional probability. I don't understand this example 1.3-4, specifically that I don't understand how they calculate P(A|B). What's the logic behind?

Answer 1

The probability that $B$ occurs multiplied by the probability that $A$ occurs given that $B$ has occured is the probability that both $A$ and $B$ occur. In the example above, $A \cap B = A$ since the intersection is only the possiblities where the sum is $3$, precisely event $A$.

$P(B) P(A | B) = P(A \cap B) = P(A)$ leads to the textbook equations.