Question: 1. We have a container with 300 red balls and 600 blue balls. We mix the balls well and choose one at random, with each ball being equally likely to be chosen. After each choice, we return the chosen ball to the container and mix again.

(a) What is the probability that the ?rst n balls chosen are all blue?

(b) Let N be the number of blue balls chosen before the ?rst red one. What is

\(\Pr \left( N=n \right)\) ? What are the mean and variance of N? Explain your answers using the formulae

\[\begin{aligned} & \sum\limits_{n=0}^{\infty }{{{x}^{n}}}=\frac{1}{1-x},\,\,\,\,\,\,\,\,\text{for }\!\!|\!\!\text{ }x|<1 \\ & \sum\limits_{n=0}^{\infty }{n{{x}^{n}}}=x\frac{d}{dx}\left( \frac{1}{1-x} \right),\,\,\,\,\,\,\,\,\text{for }\!\!|\!\!\text{ }x|<1 \\ \end{aligned}\]

(c) What is the probability that N =0 given that N ≤ 2?

(d) What is the probability that N is an even number? Count 0 as an even number.