An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop

# An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop

H
240 points

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

An urn
hansel

8.7k points

First find the probability of getting 2 white balls and 2 black balls.

$$P(\text{2 white and 2 black})=\frac{C(4,2)C(4,2)}{C(8,4)}=18/35$$

we stop when we first choose 2 white and 2 black balls, so we will make n-1 bad selections and 1 good selection which stops us at n selections or

$$P(\text{stop after n selections})=(1-p)^{n-1}p=(1-18/35)^{n-1}(18/35)$$

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