# 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?

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#### Explanation:

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Dec 7, 2017

$\frac{3}{13}$

#### 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:

$P \left(A | B\right) = \frac{P \left(A \cap B\right)}{P \left(B\right)}$

If we let A = "Drawing a face card" and B = "Drawing a spade", we can compute this by finding two values: $P \left(A \cap B\right)$, or the probability of drawing a face card which happens to also be a spade, and $P \left(B\right)$, 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 $P \left(A \cap B\right) = \frac{3}{52}$. In a similar fashion, we know there are 13 spades in a deck of 52 cards, so $P \left(B\right) = \frac{13}{52}$.

Thus:

$P \left(A | B\right) = \frac{P \left(A \cap B\right)}{P \left(B\right)} = \frac{\frac{3}{52}}{\frac{13}{52}} = \frac{3}{52} \cdot \frac{52}{13} = \frac{3}{13}$

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: $\frac{3}{13}$.

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