An office has four copying machines, and the random variable $X$ measures how many of them are in use at a particular moment in time. Suppose that $P(X = 0) = 0.08\,, \,P(X = 1) = 0.11\,,\, P(X = 2) = 0.27\,,\,$ and $P(X = 3) = 0.33$ . $\mathbf{(a)}$What is $P(X=4) ?$ $\mathbf{(b)}$ Draw a line graph of the probability mass function. $\mathbf{(c)}$ Construct and plot the cumulative distribution function.

Suppose that two fair dice are rolled and that the two numbers recorded are $\textit{multiplied}$ to obtain a final score. Construct and plot the probability mass function and the cumulative distribution function of the final score.

A fair coin is tossed three times. A player wins $\$1$ if the first toss is a head, but loses $\$1$ if the first toss is a tail. Similarly, the player wins $\$2$ if the second toss is a head, but loses $\$2$ if the second toss is a tail, and wins or loses $\$3$ according to the result of the third toss. Let the random variable $X$ be the total winnings after the three tosses (possibly a negative value if losses are incurred). $\mathbf{(a)}$ Construct the probability mass function. $\mathbf{(b)}$ Construct the cumulative distribution function. $\mathbf{(c)}$ What is the most likely value of the random variable X?

Four cards are labeled $\$1, \$2, \$3$, and $\$6$. A player pays $\$4$, selects two cards at random, and then receives the sum of the winnings indicated on the two cards. Calculate the probability mass function and the cumulative distribution function of the net winnings (that is, winnings minus the $\$4$ payment).

Two different creams are used to treat subjects with poison ivy irritation on both hands. Each subject is given a treatment of Ivy Ease on one hand while their other hand is treated with a placebo. The sample results are summarized in the table below. Use a 0.05 significance level to test the null hypothesis that the following two proportions are the same: (1) the proportion of subjects with relief on the hand treated with Ivy Ease and no relief on the hand treated with a placebo; (2) the proportion of subjects with no relief on the hand treated with Ivy Ease and relief on the hand treated with a placebo. Does the Ivy Ease treatment appear to be effective?

$$\begin{matrix} & \text{Treatment with Ivy Ease} & \text{Relief} & \text{No Relief}\\ \text{Placebo} & \text{Relief} & \text{12} & \text{8}\\ & \text{No relief} & \text{32} & \text{19}\\ \end{matrix}$$

Suppose that a deck of 25 cards contains 12 red cards. Suppose also that the 25 cards are distributed in a random manner to three players A, B, and C in such a way that player A receives 10 cards, player B receives eight cards, and player C receives seven cards. Determine the probability that player A will receive six red cards, player B will receive two red cards, and player C will receive four red cards.

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically.

$$\begin{array}{l}{y=4-x^{2}} \\ {y=2 x-1}\end{array}$$

Find all the second partial derivatives.

$$f(x,y)=x^3y^5+2x^4y$$

Draw line graphs of the probability mass functions of a $B(6, 0.5)$ distribution and a $B(6, 0.7)$ distribution. Mark the expected values of the distributions on the line graphs and calculate the standard deviations of the two distributions.

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: $(0, 4), (0, 0)$; passes through the point $(\sqrt{5},-1)$

Twenty copying jobs need to be done. If there are four copy machines, in how many ways can five jobs be assigned to each of the four machines? If an additional copier is used, in how many ways can four jobs be assigned to each of the five machines?

What is the sample space for the average score of two dice?

Consider a random variable measuring the following quantities. In each case state with reasons whether you think it more appropriate to define the random variable as discrete or as continuous. $\mathbf{(a)}$ A person’s height $\mathbf{(b)}$ A student’s course grade $\mathbf{(c)}$ The thickness of a metal plate $\mathbf{(d)}$ The purity of a chemical solution $\mathbf{(e)}$ The type of personal computer a person owns $\mathbf{(f)}$ A person’s age

A box contains 40 batteries, 5 of which have low lifetimes, 30 of which have average lifetimes, and 5 of which have high lifetimes. A consumer requires 8 batteries to run an appliance and randomly selects them all from the box. What is the probability that among the 8 batteries fitted into the consumer’s appliance, there are exactly 2 low, 4 average and 2 high lifetimes batteries?