# binomial distribution with replacing the opposite color of the items.

I'm not sure which discrete probability distribution to use for this kind of problem.

We are given two sets of balls n blue and m red. This is very similar to a binomial distribution except that every time that when drawing a ball at random, when a blue ball is drawn, 1 blue is taken out of the urn and 1 red ball is added to the urn. When a red ball is drawn the game ends.

What distribution would I use for the formula for $P \{X=k\}$ be where X is the number of blue balls drawn before the first red one is drawn?

• Everytime a blue ball is taken out a red one is added or a red ball is added no matter what colour ball is taken out? – JohnK Oct 26 '13 at 21:43
• every time a blue ball is picked, a blue one is taken out and a red one is added. – pyCthon Oct 26 '13 at 22:14

Hint: Use the identities $$P[X\geqslant k+1]=\frac{n}{n+m}\cdot\frac{n-1}{n-1+m+1}\cdots\frac{n-k}{n-k+m+k}$$ and $$P[X=k]=P[X\geqslant k]-P[X\geqslant k+1].$$