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Market makingMedium

A well-shuffled standard $52$ -card deck is laid out in order, and a contract pays $100$ dollars if the third card in the shuffled order is an ace and $0$ otherwise. You see that the first $2$ cards in the order are not aces. What is the new fair value in dollars?

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Hint

The remaining cards are equally likely to be in the third position.

Solution

After removing the first $2$ non-aces, $50$ cards remain, of which $4$ are aces.
The third card is an ace with probability $\frac{4}{50}$ .
The new fair value is $100 \times \frac{4}{50} = 8$ .

Answer: $8$

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