Discrete Probablity

Question

Dave flips a fair coin until he gets two consecutive heads and defines $X$ to be the number of flips it takes; Kevin flips another fair coin until he gets three consecutive Tails and defines $Y$ to be the number of flips it takes. What is $P(X>Y)$?

Answer

There are 12 states of the game since $X \in$ {0,1 or 2} and $Y \in$ {0,1,2 or 3}. Using joint notation "($X,Y$)", (0,3) and (1,3) are success states while (2,0), (2,1), (2,2) and (2,3) are failure states.

Let p($X,Y$) represent $P(X>Y|state\ (X,Y))$. Then we have the following state matrix:

          $Y$

         0     1      2      3

    0 p(0,0) p(0,1) p(0,2)     1

$X$  1 p(1,0) p(1,1) p(1,2)     1

    2     -1     -1     -1    -1

p(0,0) = $\frac{1}{4}*$(p(1,1)+p(0,1)+p(1,0)+p(0,0))
p(1,1) = $\frac{1}{4}*$(p(0,2)+p(0,0))
p(1,0) = $\frac{1}{4}*$(p(0,1)+p(0,0))
p(0,1) = $\frac{1}{4}*$(p(1,2)+p(1,0)+p(0,2)+p(0,0))
p(0,2) = $\frac{1}{4}*$(1+p(1,0)+1+p(0,0))
p(1,2) = $\frac{1}{4}*$(1+p(0,0))


Solving this system of equations yields p(0,0) = $\frac{361}{1699} \cong 0.212478$