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?
Solution
(a) Let 
be the probability that the player whose turn it is will hit the goal post first assuming the ball is in the bottom wicket. Then, we perform casework on the possible continuations of the game.
If Alice hits it back into the bottom wicket on her first turn, then the Queen will now have an chance of winning. Thus Alice will have a 
chance of winning. This case clearly appears with probability 
. However, if Alice hits the ball into the top wicket, then the Queen has a % chance of winning outright (and Alice's winning probability would be 0) and a % chance of returning the game to the same state as before, so Alice has probability of winning again. Thus the equation is:
![\[X=\frac{1}{2}(1-X)+\frac{1}{4}X\]](http://latex.artofproblemsolving.com/a/1/5/a1551dab0299be9130d49dfe1bbdfa6c958ecaf8.png)
![\[\Rightarrow X=\boxed{\frac{2}{5}}\]](http://latex.artofproblemsolving.com/4/0/d/40d5b375d61ac42f24922972d77c008a66e244e7.png)
(b) Define as earlier and 
as the probability for the top wicket. Then, by carefully using tree diagrams to map out possibilities (we only need to consider three turns each), we get the equations
![\[X=\frac{3}{8}X+\frac{1}{4}Y\]](http://latex.artofproblemsolving.com/c/d/5/cd5a0960309931bf8a68069f33a7136b51fb402d.png)
![\[Y=\frac{1}{2}+\frac{1}{4}X+\frac{1}{8}Y\]](http://latex.artofproblemsolving.com/f/7/4/f74643b55ffed1d6970d88a89ed5a88fae4fa632.png)
These equations imply that 
and 
after solving.
See also
| 2014 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
