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?

  • $\begingroup$ 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? $\endgroup$ – JohnK Oct 26 '13 at 21:43
  • $\begingroup$ every time a blue ball is picked, a blue one is taken out and a red one is added. $\endgroup$ – 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 $$ P[X=k]=P[X\geqslant k]-P[X\geqslant k+1]. $$


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