A and B play sudoku until A has 3 consecutive win. Find expected value of the number of matches that they played. Known that there are three cases: win, tie, lose and the probability is uniform.
I've tried using recursion approach such that: Assume f(x) is the probability that A has 3 consecutive win in x matches. We have:
$f(3) = (\frac{1}{3})^3$
$f(4) = (\frac{2}{3})(\frac{1}{3})^3$
$f(5) = (\frac{2}{3})(\frac{1}{3})^3$
$f(6) = (\frac{2}{3})(\frac{1}{3})^3$
For x > 6, assume $x = 3*n + m$ $(m = 1,2,3)$, we have:
$f(x) = (1-f(3))^{n-1}*f(m+3)$
Then, expected value is:
$E = 3*f(3)+4f(4)+5f(5)+6f(6)+\sum_{n=2}^{\infty}(3k+3)(1-f(3))^{n-1}f(6)+\sum_{n=2}^{\infty}(3k+1)(1-f(3))^{n-1}f(4)+\sum_{n=2}^{\infty}(3k+2)(1-f(3))^{n-1}f(5)=166$
The expected value is too large. Is my solution wrong?