Question

Now we have a magical urn which has N > 0 black balls and M > 0 red balls. Balls are drawn without replacement until a black one is drawn. Compute the expected number of balls drawn.

Answer 1

An urn has M red and N black balls. The balls are drawn one by one without replacement untill a black ball appears. The balls are drawn in WOR method.

P(1st ball drawn is a black ball)= N/M+N

P(the first trial resulats in a red ball and the second trial results in a black ball)= (M/M+N)*(N/M+N-1)

P(the 3rd draw results in a black ball)=

and so on...

this will continue untill all the previous balls drawn are red i.e all the M red balls are drawn then the M+1th trail must result in a black ball.

P(M+1th ball drawn is black )=

now, in order to calculate the expectation,

E(No of balls drawn before a black ball appears is)=