An urn contains 3 red and 7 black balls. Players A
and B withdraw balls from the urn consecutively
until a red ball is selected. Find the probability that
A selects the red ball. (A draws the first ball, then
B, and so on. There is no replacement of the balls
drawn.)
How do I calculate this probability?
I tried using the total probability rule without success.
I used the $P(A)=\frac{3}{10}+P(A_2\mid B_1)$ and so on, where $B_i$=Player B doesn't get a red ball.
The answer should be $0.0888$
Best Answer
Hint:
If $p(r,b)$ denotes the probability that the person drawing first selects a red ball, when there are $r$ red balls and $b$ black balls, then $p(r,0)=1$ and for $b>0$: $$p(r,b)=\frac{r}{r+b}+\frac{b}{r+b}(1-p(r,b-1))=1-\frac{b}{r+b}p(r,b-1)$$
Find $p(3,0),p(3,1),\dots,p(3,7)$ in this order.