- #1

CGandC

- 326

- 34

**Problem**:

In a box there are ##120## balls with ## X ## of them being white and ## 120 - X ## being red for random variable ##X##.

We know that ## E[ X] = 30 ##. We are taking out ## k ## balls randomly and with returning ( we return each ball we take out, so there is equal probability for each ball every time we take out a ball ), for ## k \geq 2 ##.

Let ## Y ## be the number of white balls in the sample that was taken out.

What is the probability of the first ball being white?

**Note**: This was the first question of an old exam and there were more questions that were based on it in the exam later on ( so it might be the case that we don't need to use ## k ## here. ), none of them helped in inferring the reasoning for the answer to this problem.

**The official answer was:**

## E[ \frac{X}{120}] = \frac{1}{4} ##, hence the probability of the first ball being white is ## \frac{1}{4} ##

**Question:**

I tried different things, among them being attempting to use binomial ,negative-binomial, hyper-geometric distributions in the problem, but I kept getting stuck because I don't know what ## X ## is since it is a random variable.

Then I tried getting ahead with the following equations:

## E[ X] = \sum_{i=1}^{120} x P(X = x) ## , ## E[ X] = \sum_{i=1}^{120} P(X \geq i) ## , but I was unable to proceed.

Do you have any explanation for the answer? I'm unable to retrace the steps necessary to arrive to it so I can't figure out how to arrive at the answer.

Thanks in advance for any help!