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Classify each of the following random variables as discrete or continuous. a. The time left on a parking meter b. The number of bats broken by a major league baseball team in a season c. The number of cars in a parking lot d. The total pounds of fish caught on a fishing trip e. The number of cars crossing a bridge on a given day f. The time spent by a physician examining a patient

Explain the meaning of the probability distribution of a discrete random variable. Give one example of such a probability distribution. What are the three ways to present the probability distribution of a discrete random variable?

Briefly explain the two characteristics (conditions) of the probability distribution of a discrete random variable.

The following table gives the probability distribution of a discrete random variable \(x\) $$ \begin{array}{l|lllllll} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & .11 & .19 & .28 & .15 & .12 & .09 & .06 \\ \hline \end{array} $$ Find the following probabilities. a. \(P(x=3)\) b. \(P(x \leq 2)\) c. \(P(x \geq 4)\) d. \(P(1 \leq x \leq 4)\) e. Probability that \(x\) assumes a value less than 4 f. Probability that \(x\) assumes a value greater than 2 g. Probability that \(x\) assumes a value in the interval 2 to 5

A review of emergency room records at rural Millard Fellmore Memorial Hospital was performed to determine the probability distribution of the number of patients entering the emergency room during a 1-hour period. The following table lists the distribution. $$ \begin{array}{l|ccccccc} \hline \text { Patients per hour } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Probability } & .2725 & .3543 & .2303 & .0998 & .0324 & .0084 & .0023 \\ \hline \end{array} $$ a. Graph the probability distribution. b. Determine the probability that the number of patients entering the emergency room during a randomly selected 1 -hour period is in 2 or more ii. exactly 5 iii. fewer than 3 iv. at most 1

Nathan Cheboygan, a singing gambler from northern Michigan, is famous for his loaded dice. The following table shows the probability distribution for the sum, denoted by \(x\), of the faces on a pair of Nathan's dice. $$ \begin{array}{l|ccccccccccc} \hline x & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline P(x) & .065 & .065 & .08 & .095 & .11 & .17 & .11 & .095 & .08 & .065 & .065 \\ \hline \end{array} $$ a. Draw a bar graph for this probability distribution. b. Determine the probability that the sum of the faces on a single roll of Nathan's dice is \(\begin{array}{lll}\text { i. an even number } & \text { ii. } 7 \text { or } 11 & \text { iii. } 4 \text { to } 6\end{array}\) iv. no less than 9

According to a survey, \(30 \%\) of adults are against using animals for research. Assume that this result holds true for the current population of all adults. Let \(x\) be the number of adults who are against using animals for research in a random sample of two adults. Obtain the probability distribution of \(x\). Draw a tree diagram for this problem.

In a group of 20 athletes, 6 have used performance-enhancing drugs that are illegal. Suppose that 2 athletes are randomly selected from this group. Let \(x\) denote the number of athletes in this sample who have used such illegal drugs. Write the probability distribution of \(x\). You may draw a tree diagram and use that to write the probability distribution. (Hint: Note that the selections are made without replacement from a small population. Hence, the probabilities of outcomes do not remain constant for each selection.)

Let \(x\) be the number of magazines a person reads every week. Based on a sample survey of adults, the following probability distribution table was prepared. $$ \begin{array}{l|cccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(x) & .36 & .24 & .18 & .10 & .07 & .05 \\ \hline \end{array} $$ Find the mean and standard deviation of \(x\).

A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are $$\$ 10$$ million from the office building, $$\$ 5$$ million from the theater, and $$\$ 2$$ million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are \(.15, .30, .45\), and 10, respectively. Let \(x\) be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of \(x\). Find the mean and standard deviation of \(x\). Give a brief interpretation of the values of the mean and standard deviation.