Question

Problems based on negative hypergeometric distribution

Answer 1

Hey,

Suppose we've given an urn which contains R red and W white balls. These balls are drawn randomly from the urn and are not placed back. Let

X:= number of attempts, before we've drawn at least r≤R red balls

Y:= number of red balls after n−1 attempts

I want to calculate Pr(X=n).

Suppose we've already drawn n−1 balls from the urn and received r−1 red balls. The probability of this to happen is given by

Pr(Y=r−1)=h(r−1|R+W,R,n−1):=(Rr−1)(Wn−r)(R+Wn−1)

where h denotes the hypergeometric distribution. The probability that now another red ball is drawn is given by

Pr(X=n)=Pr(Y=r−1)h(1|R+W−(n−1),R−(r−1),1)=(Rr−1)(Wn−r)(R+Wn−1)R−(r−1)R+W−(n−1)

While I think that's correct, I would really like to get a more compact form of that. After some research on the internet, I found out, that I should be able to receive

Pr(X=n)=…=(r−1n−1)(R+W−(r−1)R−(n−1))(R+WR)

HOPE IT HELPS.... :)