There are $n$ urns each having $a$ white and $b$ black balls. One ball is taken from urn 1 and is transferred to urn 2. Then one ball is taken from urn 2 and transferred to urn 3 and so on. Find the probability that the ball drawn from $n$th urn is white.
I get the intuition that the answer should be $\frac{a}{a+b}$, but I'm unable to prove it.
Best Answer
When you go from urn 1 to urn 2, the white probability in urn 2 is now $\frac{a+\frac{a}{a+b}}{a+b+1} = \frac{(a+b)a+a}{(a+b+1)(a+b)}=\frac{a}{a+b}$ Therefore as you continue, the probabilities in each urn remain the same after each transfer, leading to your final answer.