I am solving problems from the book Introduction to probability by Dimitri P. Bertsekas and John N. Tsitsiklis and there came a series of problems in which you have to develop recursive formulas for probability.
Here is the problem I have the trouble with:
Two players take turns removing a ball from a jar that initially contains m white and k black balls. The first player to remove a white ball wins. Develop a recursive formula that allows the convenient computation of the probability that the starting player wins.
Solution:
$$p(m,k)=\frac{m}{m+k}+\frac{k}{m+k}\left(1-p(m,k-1\right))$$
My question is:
- What is the logic behind the $$(1-p(m,k-1))$$
- Are there any useful resources where I can learn about recursion in probability?
Started with $m$ white balls and $k$ black balls.
Probability that you get a white ball at the first draw is $\frac{m}{m+k}$
Suppose the first draw is not white, that happens with probability $\frac{k}{m+k}$, then we have to multiply with the probability that the second player lose. Now the second player starts the new game with $m$ white balls and $k-1$ black balls, the probability that he lose is $1-p(m,k-1)$
Hence the formula.