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.
Let p be the chance of cutting a spade and q the chance of not cutting a spade from a pack of 52 cards.
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
Similarly B' s chance =(qp+q4p+q7p+⋯)
and C's chance = 34 of B's chance=34.1237=937