suppose you have red and blue balls in a jar and you pick one at a time,
- if its red you replace it, additionally you do another draw and throw whichever ball comes up.
- if its blue you throw it away
Assuming R red and B blue balls, whats probability that last ball would be blue
Hint: After each iteration you have one less ball, and if you have $n$ total balls at a given time, $B$ blue balls and $n-B$ red balls, then you can write down a recurrence for the probability of the last ball being blue assuming you have $B$ blue out of $n$ total balls, let's denote the probability $p(n,B)$, in terms of $p(n-1,B)$ and $p(n-1,B-1)$ after you apply your random drawing process one time. You can try to find a pattern and prove it with this recursion, or just attack the recursion directly if you know how.