Chapter 3, Problem 3.7TE

a. An urn contains n white and m black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability $\frac{n}{\left(n+m\right)}$, they are all white. Hint: Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn.

b. A pond contains 3 distinct species of fish, which we will call the Red, Blue, and Green fish. There are r Red, b Blue, and g Green fish. Suppose that the fish are removed from the pond in a random order. (That is. each selection is equally likely to be any of the remaining fish.) What is the probability that the Red fish are the first species to become extinct In the pond?

Hint: Write $\text{P}\left\{\text{R}\right\}=\text{ P}\left\{\text{RBG}\right\}+\text{ P}\left\{\text{RCB}\right\}$, and compute the probabilities on the right by first conditioning on the last species to be removed.