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Consecutive aces

ProbabilityEasy

A well-shuffled standard deck has $52$ cards, including $4$ aces. You draw $2$ cards without replacement. What is the probability that both cards are aces? Give a fraction in lowest terms.

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Hint

Multiply the chance for the first card by the conditional chance for the second card.

Solution

The first card is an ace with probability $\frac{4}{52}$ .
After that, $3$ aces remain among $51$ cards, so the second card is an ace with probability $\frac{3}{51}$ .
Multiply: $\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652}$ .
Reduce the fraction to lowest terms; the probability is $\frac{1}{221}$ .

Answer: $\frac{1}{221}$

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