2.6.5 problem 64

Let $R_i$, $G_i$, and $B_i$ be the events that the $i$-th drawn ball is red, green or blue respectively. Let $A$ be the event that a green ball is drawn before a blue ball.

(a)

Note that if a red ball is drawn, it is placed back, as if the experiment never happened. Draws continue until a green or a blue ball is drawn. The red balls are irrelevant in the experiment. Thus, the problem reduces to removing all the red balls, and finding the probability of the first, randomly drawn ball being green.

$$\begin{array}{cccc}P\left(A\right)& =P\left(R_1\right)P\left(A\right|R_1)+P(R_1^c\left)P\right(A\left|R_1^c\right)& & \\ & =\mathrm{rP}\left(A\right)+(g+b)\frac{g}{g+b}& & \\ & =\mathrm{rP}\left(A\right)+g& & \end{array}$$

Thus,

$$P\left(A\right)=\frac{g}{1-r}=\frac{g}{g+b}.$$

(b)

We are interested in draws in which the first ball is green. Each completed sequence of $g+b+r$ draws is equally likely. Since the red balls are once again irrelevant, we focus on the $g+b$ draws of green or blue balls.

Thus,

$$P\left(A\right)=\frac{\left(\frac{g+b-1}{g-1}\right)}{\left(\frac{g+b}{g}\right)}=\frac{g}{g+b}.$$

(c)

Let $A_{i,j}$ be the event that type $i$ occurs before type $j$. Generalizing part $a$, we get $$P\left(A_{i,j}\right)=\frac{p_i}{p_i+p_j}.$$