Okay, here’s a practical example that illustrates how conditional probability works. Suppose you have an ordinary 52-card deck. If you pick a card at random from the deck, the probability that it’s a heart is ¼ = 0.25. Suppose further that you can’t see the card you’ve drawn but your son tells you it’s a red card. What is the probability that it’s a heart now? (The answer is 13/26 = ½ = 0.5 because there are 26 red cards in a deck, of which 13 are hearts.)

So in this case we can phrase the probabilities as follows:

P(A|B) = P(A & B)/P(B)

(Read: “The probability of A, given that B, is the joint probability of A **and** B, divided by the probability of B.”)

We let A = “Draw a heart” and B = “Draw a red card”. Then P(A & B) is the probability that it’s a red card and it’s a heart. But if it’s a heart, it’s automatically a red card, so P(A & B) = P(A) = 13/52 = ¼ = 0.25, as before. P(B) is the probability it’s a red card, which is 26/52 = ½ = 0.5. Plugging our values into the above conditional probability formula, we get P(A|B) = P(A & B)/P(B) = ¼/½ = ½ = 0.5.

The gist of it is that in “A, given that B”, knowledge of the occurrence of event B in some way affects the sample space of event A — normally, the sample space of event A is a subset of the sample space of event B.