Probability sampling question

Question

A bag contains k (inedible) regular buttons and one chocolate button. Two

players in turn take a button at random without replacement from the bag, until the

chocolate button is drawn. The player drawing the chocolate button is the winner,

and is permitted to eat the delicious prize button. If player A goes rst, what's the

probability that he wins?

Answer 1

Suppose that the 2 players draw all the buttons. Player A has a buttons and Player B has b buttons such that a+b = k. Now the chocolate button can be any button. Player A's chance is a/k. If k is even player A has a 1/2 chance. Otherwise player A has a (k+1)/(2k) chance.