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From a standard deck of 52 cards, what is the probability that a randomly drawn card is a King, on condition that the card drawn is a Heart?

I used the conditional probability formula and got:

Probability that the card is a King AND a Heart: $\frac{1}{52}$

Probability that the card is a Heart: $\frac{13}{52}$

So: $\frac{\frac{1}{52}}{\frac{13}{52}} = \frac{1}{13}$.

Is this correct?

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    $\begingroup$ yup. as an extra observation you can note that this is exactly the probability of drawing a king without any information on the color of the card drawn $\endgroup$
    – mm-aops
    Commented Jun 18, 2014 at 22:35

2 Answers 2

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Yes. There are thirteen hearts, and only one of them is a king.

$$\Pr(K \mid \heartsuit) = \frac{\Pr(K \cap \heartsuit)}{\Pr(\heartsuit)} = \frac{^1\!/_{52}}{^{13}\!/_{52}}=\frac{1}{13}$$

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  • $\begingroup$ +1 for correctness and for showing that "heart" can be a math symbol. :) $\endgroup$
    – StumpyLeg
    Commented Jun 18, 2014 at 22:44
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We want to work out $P(king|heart)$.

$P(heart)=1/4$

$P(king \& heart)=1/52$

Conditional probability formula: $P(A|B)=P(A \& B)/P(B)$.

So substituting into this formula we get:

$P(king | heart) = P(king \& heart) / P(heart) = (1/52)/(1/4) = 4/52 = 1/13$ as required.

So yes, you are correct.

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