Q.7.36 - A First Course in Probability

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. 

Step-by-Step Solution

Answer

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

Step 1: 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. 

Step 2: 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 $\left(1\right)$.

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

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

Step 3: Final answer

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