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Question

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


  1. 1637,1237,937
  2. 1621,421,121
  3. 937,1237,1637
  4. 121,421,1621


Solution

The correct option is A 1637,1237,937

Let p be the chance of cutting a spade and q the chance of not cutting a spade from a pack of 52 cards.
Then  p=13C152C1=14q=114=34.
Now A will win a prize if he cuts spade at 1st, 4th, 7th, 10th turns, etc. Note that  A will get a second chance if A,B,C all fail to cut a spade once and then A cuts a spade at the 4th turn. Similarly he will cut a spade at the 7th turn when A,B,C fail to cut spade twice etc.
Hence A's chance of winning the prize 
=p+q3p+q6p+q6p+q9p+=p1q3=141(34)3=1637.

Similarly B' s chance =(qp+q4p+q7p+)
=q(p+q3p+q6p+)
34.1637=1237
and C's chance = 34 of B's chance=34.1237=937

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