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Working out Choose a person aged 19 to 25 years at random and ask, "In the past seven days, how many times did you go to an exercise or fitness center or work out?" Call the response \(Y\) for short. Based on a large sample survey, here is a probability model for the answer you will get: \({ }^{8}\) $$ \begin{array}{lcccccccc} \hline \text { Days: } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \text { Probability: } & 0.68 & 0.05 & 0.07 & 0.08 & 0.05 & 0.04 & 0.01 & 0.02 \\\ \hline \end{array} $$ (a) Show that this is a legitimate probability distribution. (b) Make a histogram of the probability distribution. Describe what you see. (c) Describe the event \(Y<7\) in words. What is \(P(Y<7) ?\) (d) Express the event "worked out at least once" in terms of \(Y\). What is the probability of this event?

Ace! Professional tennis player Rafael Nadal hits the ball extremely hard. His first-serve speeds follow a Normal distribution with mean 115 miles per hour (mph) and standard deviation 6 mph. Choose one of Nadal's first serves at random. Let \(Y=\) its speed, measured in miles per hour. (a) Find \(P(Y>120)\) and interpret the result. (b) What is \(P(Y \geq 120)\) ? Explain. (c) Find the value of \(c\) such that \(P(Y \leq c)=0.15 .\) Show your work.

Exercises 27 to 29 refer to the following setting. Choose an American household at random and let the random variable \(X\) be the number of cars (including \(\mathrm{SUVs}\) and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: \begin{tabular}{lcccccc} \hline Number of cars \(X:\) & 0 & 1 & 2 & 3 & 4 & 5 \\ Probability: & 0.09 & 0.36 & 0.35 & 0.13 & 0.05 & 0.02 \\ \hline \end{tabular} 27\. What's the expected number of cars in a randomly selected American household? (a) 1.00 (b) 1.75 (c) 1.84 (d) 2.00 (e) 2.50

The Tri-State Pick 3 Most states and Canadian provinces have government- sponsored lotteries. Here is a simple lottery wager, from the Tri-State Pick 3 game that New Hampshire shares with Maine and Vermont. You choose a number with 3 digits from 0 to \(9 ;\) the state chooses a three-digit winning number at random and pays you \(\$ 500\) if your number is chosen. Because there are 1000 numbers with three digits, you have probability \(1 / 1000\) of winning. Taking \(X\) to be the amount your ticket pays you, the probability distribution of \(X\) is $$ \begin{array}{lcc} \hline \text { Payoff: } & \$ 0 & \$ 500 \\ \text { Probability: } & 0.999 & 0.001 \\ \hline \end{array} $$ (a) Show that the mean and standard deviation of \(X\) are $$ \mu_{X}=\$ 0.50 \text { and } \sigma_{X}=\$ 15.80 $$ (b) If you buy a Pick 3 ticket, your winnings are \(W=X-1\), because it costs \(\$ 1\) to play. Find the mean and standard deviation of \(W\). Interpret each of these values in context.

Rainy days Imagine that we randomly select a day from the past 10 years. Let \(X\) be the recorded rainfall on this date at the airport in Orlando, Florida, and \(Y\) be the recorded rainfall on this date at Disney World just outside Orlando. Suppose that you know the means \(\mu_{X}\) and \(\mu_{Y}\) and the variances \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\) of both variables. (a) Is it reasonable to take the mean of the total rainfall \(X+Y\) to be \(\mu_{X}+\mu_{Y} ?\) Explain your answer. (b) Is it reasonable to take the variance of the total rainfall to be \(\sigma_{\mathrm{X}}^{2}+\sigma_{\mathrm{Y}}^{2}\) ? Explain your answer.

determine whether the given random variable has a binomial distribution. Justify your answer. Sowing seeds Seed Depot advertises that its new flower seeds have an \(85 \%\) chance of germinating (growing). Suppose that the company's claim is true. Judy gets a packet with 20 randomly selected new flower seeds from Seed Depot and plants them in her garden. Let \(X=\) the number of seeds that germinate.

Elk Biologists estimate that a baby elk has a \(44 \%\) chance of surviving to adulthood. Assume this estimate is correct. Suppose researchers choose 7 baby elk at random to monitor. Let \(X=\) the number who survive to adulthood. Use the binomial probability formula to find \(P(X=4) .\) Interpret this result in context.

Random digit dialing When an opinion poll calls a residential telephone number at random, there is only a \(20 \%\) chance that the call reaches a live person. You watch the random digit dialing machine make 15 calls. Let \(X=\) the number of calls that reach a live person. (a) Find and interpret \(\mu_{X}\) (b) Find and interpret \(\sigma_{X}\)

"Large Counts condition To use a Normal distribution to approximate binomial probabilities, why do we require that both \(n p\) and \(n(1-p)\) be at least \(10 ?\)

Joe reads that 1 out of 4 eggs contains salmonella bacteria. So he never uses more than 3 eggs in cooking. If eggs do or don't contain salmonella independently of each other, the number of contaminated eggs when Joe uses 3 chosen at random has the following distribution: (a) binomial; \(n=4\) and \(p=1 / 4\) (b) binomial; \(n=3\) and \(p=1 / 4\) (c) binomial; \(n=3\) and \(p=1 / 3\) (d) geometric; \(p=1 / 4\) (e) geometric; \(p=1 / 3\)