2014 UNCO Math Contest II Problems/Problem 8
Problem
In the Queen’s croquet, a game begins with the ball at the bottom
wicket. All players hit the same ball. Each player hits the ball from
the place the previous player has left it. When the ball is hit from
the bottom wicket, it has a 
% chance of going to the top wicket
and a % chance of staying at the bottom wicket. When hit from
the top wicket, it has a % chance of hitting the goal post and a
50% chance of returning to the bottom wicket.
(a) If Alice makes the first hit and alternates hits with the Queen, what is the probability that Alice is the first player to hit the goal post with the ball?
(b) Suppose Alice, the King, and the Queen take turns hitting the ball, with Alice playing first. Now what is the probability that Alice is the first player to hit the goal post with the ball?
