**Question 1:** It is easiest to prove this by induction. If $r = 0$ (there are no red balls) and $b \geq 1$, then clearly the first ball will be blue. That is, the number of balls $N$ that we need to draw in order to draw a blue ball has expected value $E(N) = 1$. Note that we can write this as

$$
E(N) = \frac{b+1}{b+1} = 1
$$

in accordance with the desired formula. Now, consider any other value of $r$, and $b \geq 1$ still. Suppose that we already have shown that for $r-1$ red balls and $b$ blue balls, the expected number of balls until we draw a blue ball is $\frac{b+r}{b+1}$, in accordance with the desired formula. We now draw a ball from an urn with $r$ red balls and $b$ blue balls. With probability $\frac{b}{b+r}$, the first ball is blue, and $N = 1$. With probability $\frac{r}{b+r}$, the first ball is red, and we now have an urn with $r-1$ red balls and $b$ blue balls. By the premise, we already know what the expected number of additional balls that need to be drawn in order to produce a blue ball; it is $\frac{b+r}{b+1}$. Therefore, for the urn with $r$ red balls and $b$ blue balls, we have

$$
\begin{align}
E(N) & = \frac{b}{b+r} \cdot 1
+ \frac{r}{b+r} \cdot \left(1+\frac{b+r}{b+1}\right) \\
& = \frac{b}{b+r} + \frac{r}{b+r} + \frac{r}{b+1} \\
& = 1 + \frac{r}{b+1} = \frac{b+r+1}{b+1}
\end{align}
$$

and we are done.

**Question 2:** By symmetry, the expected number of remaining balls left when only balls of one color (either red or blue) remain is equal to the number of balls of the same color we draw *at the beginning*. (To see this, draw all the balls out from the urn, and lay them out on a table, in sequence. The sequence from left to right is exactly as probable as the sequence from right to left.)

With probability $\frac{r}{b+r}$, the first ball is red. We are then left with $r-1$ red balls and $b$ blue balls, and we want to know how many balls we draw *before* (not until) we draw the first blue ball. This is one less than the result from Question 1. To this we have to add the first red ball. So when the first ball is red, our answer is

$$
N_{red} = \frac{b+r}{b+1}-1+1 = \frac{b+r}{b+1}
$$

With probability $\frac{b}{b+r}$, the first ball is blue, and by symmetry, the number of consecutive blue balls at the start is

$$
N_{blue} = \frac{b+r}{r+1}
$$

Thus, the expected number of consecutive balls drawn of the same color (which is also the answer to the original Question 2) is

$$
E(N) = \frac{r}{b+r} \cdot N_{red} + \frac{b}{b+r} \cdot N_{blue}
= \frac{r}{b+1} + \frac{b}{r+1}
$$

## Best Answer

$$P(A|B) = \frac{P(A\cap B)}{P(B)}$$

$$= \frac{P(2 \,red \,balls\, AND \,even\, number\, of\, red\, balls\,)}{P(0\, red\, balls\, OR\, 2\, red\, balls)}$$

$$= \frac{P(2\,red\,balls)}{P(0\, red\, balls\, OR\, 2\, red\, balls)}$$

$$=\frac{\frac{\binom{7}{2}\binom{9}{1}}{\binom{16}{3}}}{\frac{\binom{9}{3}+\binom{7}{2}\binom{9}{1}}{\binom{16}{3}}}$$

$$=\frac{\binom{7}{2}\binom{9}{1}}{\binom{9}{3}+\binom{7}{2}\binom{9}{1}}$$