Conditional Card Probability

P(B|A) with playing card examples

P(Red | Heart)

Question

You draw a card from a deck and are told it is a Heart. What is the probability that it is a red card?

♥

Hearts

13 cards

Red cards are Hearts and Diamonds:

♥

Hearts

♦

Diamonds

Hearts are a subset of red cards. Every Heart IS a red card!

Since every Heart is red:

Answer

This is a trivial case of conditional probability. When B is a subset of A (every Heart is Red), then P(A|B) = 1.

Red Cards

Hearts

Hearts are entirely contained within Red cards.

The reverse question has a different answer:

If we know it's red, it could be Heart or Diamond (50% each).