Question

A jar contains 4 white and 4 black marbles. We randomly choose 4 marbles. If 2 of them are white and 2 are black, we stop. If not, we replace the marbles in the jar and again randomly select 4 marbles. This continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly n selections?

Answer 1

Find the probability of success in a single trial and then think about the nature of the problem (when do we stop).

Explanation:

Observe that in the single trial, we have (8 4) possibilities of choosing our set of balls. If we have chosen two white balls and two black balls, the probability of doing that is simply

p=(4 2)*(4 2)/(8 4)

This is well know Hyper geometric distribution. Now, define random variable X that marks the number of trials that have been needed to obtain the right combination (two white and two black balls). From the nature of the problem, observe that X has Geometric distribution with parameter p that has been calculated above. Hence

P(X = n) = (1— p)^n-1 *( p )

Find the probability of success in a single trial and then think about the nature of the problem (when do we stop).