A standard deck has 52 cards in 4 suits of 13 (ace, 2 to 10, jack, queen, king); hearts and diamonds are red, clubs and spades are black, and the jack, queen and king are the face cards. Two cards are drawn one after the other without replacement. What is the probability that both are aces?

\frac{1}{442}

Correct answerYour answer

\frac{1}{221}

Correct answerYour answer

\frac{1}{169}

Correct answerYour answer

\frac{1}{26}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

First card an ace: $\frac{4}{52}$ . Second card an ace, with 3 aces left in 51 cards: $\frac{3}{51}$ .
P(both aces) $= \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$ .

Worth noting

$\frac{1}{169} = \left(\frac{4}{52}\right)^2$ is WITH replacement. $\frac{1}{442}$ divides ${}^{4}C_{2} = 6$ by the ordered count $52 \times 51$ , mixing combinations with ordered draws.

Concept: Two dependent card draws
Typical timeTypical: 70 s
Your timeYours: —

The rule this testsCoins, dice and cards

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Ten questions from Counting and Basic Probability, at your own pace.

A standard deck has 52 cards in 4 suits of 13 (ace, 2 to 10, jack, queen, king); hearts and diamonds are red, clubs and spades are black, and the jack, queen and king are the face cards. Two cards are drawn one after the other without replacement. What is the probability that both are aces?

Explanation