Question

A bag contains $3$ white chips and $3$ red chips. you repeatedly draw a chip at random from the bag. if it's white, you set it aside; if it's red, you put it back in the bag. after removing all $3$ white chips, you stop. what is the expected number of times you will draw from the bag?

Answer 1

Answer:

8.5

Step-by-step explanation:

We can divide this experiment in three parts.

• Before removing any white chip
• After ramoving one white chip, and before removing two
• After removing two white chips

For the first experiment, for each extraction we will always have 3 white chips and 3 red chips, becuase if we extract a red chip, then we put it back in the bag, and if it is a white chip, then the experiment ends there. The probability of taking out a white chip is 1/2.

For the second experiment, we will have always 2 white chips, and 3 red chips. So the probability of success is 2/5 = 0.4

For the third experiment, we will have always 1 white chip and 3 red chips, so the probability of success if 1/4 = 0.25.

We want to know how many extractions we need for each experiment until we pick a white chip. Note that each experiment is independent of each other, and each one has geometric distribution, the first one with probability of success 0.5, the second one with probability of success 0.4, and the third one with p = 0.25.

The total experiment, X, is the sum of this random variables, so the expected value of the total experimet is the sum of the expected value from each of its parts, lets call them, X₁, X₂ and X₃. Thus,

$E(X) = E(X_1) + E(X_2) + E(X_3) = \frac{1}{0.5}+\frac{1}{0.4} +\frac{1}{0.25} = 2 + 2.5 + 4 = 8.5$

The expected number of draws is 8.5.

I hope i could help you!