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In a jar there are two balls: black and white.

We draw one ball every time, chacking his color and return it to the jar with another ball of the same color.

The test is ended after we pick black ball for the first time.

Let $X$ be the number of steps until the end of the test (When a black ball has been picked).

I want to find the distribution

of $X$ and also $E[X]$ which represents the expectation of $X$.

I wanted to use geometric distribution, I just didnt know how to handel with the increasing number of balls.

Thank you

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    It is worth looking at the first few cases to spot a pattern.2011-04-24

2 Answers 2

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Forget about named distributions. Note that after $n$ steps in which the ball picked was white, there are $n+1$ white balls and one black ball. So what is the (conditional) probability that the next ball picked is black?

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What is the probability that the test ends in the first step i.e. what is the probability of picking the black ball in the first attempt?

What is the probability that the test ends in the second step i.e. what is the probability of picking the white ball in the first attempt and the black ball in the second attempt? (Note that at the second chance to pick we have $2$ white balls and one black ball)

In general, What is the probability that the test ends in the $n^{th}$ step i.e. what is the probability of picking a white ball in the first $n-1$ attempts and the black ball in the $n^{th}$ attempt? (Note that at the $n^{th}$ chance to pick we have $n$ white balls and one black ball)