There are $n$ types of balls in an urn; $a$ balls of type $1$ , $b$ balls of type $2$ , $c$ balls of type $3$ and so on. Now balls are drawn until a ball of type $1$ is obtained with condition that if a ball of type $i$ is drawn at any moment ( $i \neq 1$ ) then all balls of type $i$ will be removed from the urn. What is the expected number of draws done? (In one draw, all balls of that particular type are removed)
My approach for $n = 3$ is like:
expectation value = $a/(a+b+c)+2(ba/(a+b+c)(a+c)+ca/(a+b+c)(a+b))+3(bca/(a+b+c)(a+c)a+cba/(a+b+c)(a+b)a)$
But it's messy , is there any general form for n type of balls ?
Prob(Balls of type $2$ are removed)=$b/(a+b)$.
Average number of draws = $$1+\sum_i\frac{b_i}{a+b_i}$$