Question

One ball at a time is randomly selected from an urn containing a white and b black balls until all of the remaining balls are of the same color. Let Ma,b denote the expected number of balls left in the urn when the experiment ends. Compute a recursive formula for Ma,b and solve when a = 3 and b = 5.

Short answer

The solution for the equation will be$\frac{3}{8}{M}_{2,5}+\frac{5}{8}{M}_{3,4}$.

Step-by-step solution

Given information

One ball at a time is randomly selected from an urn containing a white and b black balls until all of the remaining balls are of the same color.

Solution

We need to calculate a recursive formula for $M_{a,b}$and solve when $a=3$and $b=5$.

$M_{a,b}=$the expected number of balls left in the urn when the experiment ends.

recursive formula will be,

$M_{a,b}=\frac{a}{a+b}{M}_{a-1,b}+\frac{b}{a+b}{M}_{a,b-1},a,b>0$

Substitute $3$for $a$and $5$for $b$in equation $\left(1\right)$.

$M_{3,5}=\frac{3}{3+5}{M}_{3-1,5}+\frac{5}{3+5}{M}_{3,5-1}$

$=\frac{3}{8}{M}_{2,5}+\frac{5}{8}{M}_{3,4}$

Final answer 

The solution for the equation will be$\frac{3}{8}{M}_{2,5}+\frac{5}{8}{M}_{3,4}$.