How to solve the recurrence relation for $n >=m$: $$P_{n,m}=\frac{n}{n+m}P_{n-1, m} + \frac{m}{m+n}P_{n,m-1}$$ $$P_{11}=\frac{1}{2}; P_{i,0}=1 \forall i > 0; P_{i,j}=0 \forall i<j$$
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Hint: Have you tried calculating the first few numbers. There is a interesting pattern. $$\begin{array}{c|cccc} &0&1&2&3\\ \hline 1&\frac{2}{2} & \frac{3}{3} & \frac{4}{4}& \frac{5}{5}\\ 2&\frac{1}{2} & \frac{2}{3} & \frac{3}{4} & \ddots\\ 3&0 & \frac{1}{3} & \frac{2}{4} & \ddots\\ 4&0 & 0 & \frac{1}{4} & \ddots\\ \end{array}$$
It looks like, that your formular for $P_{n,m}$ is $$P_{n,m} = \left\{\begin{array}{c,l}0\quad &\text{if }m>n\\ \frac{n+1-m}{n+1}\quad &\text{else}\end{array}\right.$$
Try to proof that by using induction over $n$ and over $m$.
Jakube
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Your indexing is off slightly: the lefthand column should be $0,1,2,3$. – Brian M. Scott Mar 09 '13 at 17:06