From an urn containing a white and b black balls, balls are drawn one
by one at random according to the following rules:
(i) at any drawing, if the ball drawn is white, then it is returned to the urn,
(ii) if it is black, it is replaced by a white ball (from another collection of
balls).
After n such operations, a ball is drawn from the urn. Find the probability
that it will be white.
my Approach:
Let w(k) denote that ball from kth selection is white,k = 1,2,…,n
suppose p_k = P(w(k)),we have to find p_n = P(w(n))..
i know surely that,we have to obtain Recurrence relation,but don't know how to manipulate p_n.
Correct Answer:$1-($($b/(a+b)$)$(1-1/(a+b))^{n}$)
Best Answer
There are $w+b$ balls. Number them from $1$ to $w+b$. When a black ball is replaced, give the new white ball the same number.
For the ball to be black on the $k^{th}$ draw, it must be the first time that number was drawn. The number of ways to do that is $(w+b-1)^{k-1}$.
Can you finish the question?