30 Jun
2014
30 Jun
'14
3:31 p.m.
On 30/06/2014 20:50, Bernie Cosell wrote:
You have a jar with N *each* of C colors of marbles [that is, you start with the same number of each color, N*C marbles in the jar]. You start removing marbles [without replacement]. What is the expected number of marbles you have to remove to have a 50% chance of having at least one of each color.
Inclusion-exclusion: Pr(at least one of each colour) = 1 - sum{i} Pr(none of colour i) + sum{i,j} Pr(none of colour i,j) - sum{i,j,k} Pr(none of colour i,j,k) + ... = sum (-1)^r (C choose r) ((C-r)N choose M) / (CN choose M) if M is the number of marbles you take. This doesn't seem like the sort of thing that would have an obvious closed form but I'm not an expert on such sums. -- g