Multiple choice

A, B, C in order cut a pack of cards replacing them after each cut; on condition that the first who cuts a spade shall win a prize; find their respective chances.

  1. $\displaystyle \frac{15}{37},\frac{12}{37},\frac{10}{37}$
  2. $\displaystyle \frac{16}{37},\frac{11}{37},\frac{10}{37}$
  3. $\displaystyle \frac{16}{37},\frac{12}{37},\frac{9}{37}$
  4. $\displaystyle \frac{14}{37},\frac{13}{37},\frac{10}{37}$
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C Correct answer
AI explanation

Let p be the probability of cutting a spade, which is 1/4, making the probability of failure q equal to 3/4. Since A starts, A's probability of winning is p divided by (1 minus q cubed), which is (1/4) / (1 - 27/64) = (1/4) / (37/64) = 16/37. B's probability of winning is q times A's probability, giving (3/4) x (16/37) = 12/37, and C's probability is q times B's probability, giving (3/4) x (12/37) = 9/37.

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