First ace on the fifth card

Question

A small deck holds 3 aces and 9 non-aces, 12 cards in total. You deal cards one at a time, without replacement. What is the probability that the first ace appears on exactly the fifth card?

Accepted Answer

"First ace on the fifth card" means the first four cards are all non-aces and the fifth is an ace. Chain the draws, shrinking the deck. Cards 1 to 4 all non-ace. Non-aces go 9 of 12, 8 of 11, 7 of 10, 6 of 9: \frac{9}{12}	imes\frac{8}{11}	imes\frac{7}{10}	imes\frac{6}{9}. Card 5 is an ace. After 4 non-aces leave, all 3 aces remain out of the 8 cards left: \frac{3}{8}. Multiply everything: \frac{9}{12}	imes\frac{8}{11}	imes\frac{7}{10}	imes\frac{6}{9}	imes\frac{3}{8}=\frac{9072}{95040}=\frac{21}{220}\approx 0.095. Takeaway. The longer the wait, the more failure draws you chain before th…

First ace on the fifth card

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