Question

What is the conditional probability that a card drawn at random from a pack of 52 cards is a face card, given that the drawn card is a spade?

Answer 1

`3/13`

Explanation:

The formula for the conditional probability of event A happening given that it's known event B already happened is given by the formula:

`P(A | B) = (P(A nn B))/(P(B))`

If we let A = "Drawing a face card" and B = "Drawing a spade", we can compute this by finding two values: `P(A nn B)`, or the probability of drawing a face card which happens to also be a spade, and `P(B)`, or the probability of drawing a spade.

Since there are three face cards (Jack, Queen, and King) in the spades suit, and 52 total possible cards, the `P(A nn B) = 3/52`. In a similar fashion, we know there are 13 spades in a deck of 52 cards, so `P(B) = 13/52`.

Thus:

`P(A | B) = (P(A nn B))/(P(B)) = (3/52)/(13/52) = 3/52 * 52/13 = 3/13`

Alternative

It's easier to do this when you recognize that knowing the drawn card was a spade has "collapsed" the set S of possibilities down to just 13 cards (the spades). Of those 13, only 3 are face cards. Thus: `3/13`.