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Ace pair contract

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From a well-shuffled standard $52$ -card deck, $2$ cards are dealt one after another, without replacement. A contract pays $100$ dollars if both cards are aces and $0$ otherwise. What is its fair value in dollars? Give an exact fraction.

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Hint

Multiply the conditional probabilities of the successive draws.

Solution

The first card is an ace with probability $\frac{4}{52}$ .
Given that, the second is an ace with probability $\frac{3}{51}$ .
The probability of a pair of aces is $\frac{12}{2652} = \frac{1}{221}$ , so the fair value is $\frac{100}{221}$ .

Answer: $\frac{100}{221}$

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