Exercises 3.12: 1, 5(a), 6, 8, 10

Also:

For each of the following, find the probability mass function for the given random variable:

1. Suppose 4 chips are drawn, without replacement, from an urn with 3 red chips and 2 blue chips. Let $X$ be the number of red chips drawn.
2. $X$ is the number of heads in four tosses of a fair coin.
3. $Y$ is number of aces when two cards are drawn a random from a standard deck of 52 cards.

Answers:

1. $p_X(x) = \begin{cases}\frac{3}{5},& \text{if } x = 2, \\ \frac{2}{5},& \text{if } x = 3, \\ 0,& \text{otherwise}.\end{cases}$
2. $p_X(x) = \begin{cases}\frac{1}{16},& \text{if } x = 0, \\ \frac{1}{4},& \text{if } x = 1, \\ \frac{3}{8},& \text{if } x = 2, \\ \frac{1}{4},& \text{if } x = 3, \\ \frac{1}{16},& \text{if } x = 4, \\ 0,& \text{otherwise}. \end{cases}$
3. $p_Y(y) = \begin{cases}\frac{188}{221},& \text{if } y = 0, \\ \frac{32}{221},& \text{if } y = 1, \\ \frac{1}{221},& \text{if } y = 2, \\ 0,& \text{otherwise}.\end{cases}$