In $2015$, the U.S. population was about $321$ million. The overall birth rate was $12.5$ births per $1000$ people, and the overall death rate was $8.4$ deaths per $1000$ people. a. Approximately how many births were there in the United States in $2015$? b. About how many deaths were there in the United States in that year? c. Based on births and deaths alone (i.e., not counting immigration), about how much did the U.S. population increase during $2015$? d. Suppose that during $2015$ the U.S. population actually increased by $3.0$ million. Based on this fact and your results from part (c), estimate how many people immigrated to the United States. What proportion of the overall population growth was due to immigration?

Use the graphs in Figure $7.14$ to answer the following question. Suppose that a life insurance company insures $1$ million $50$ -year-old people in a given year. (Assume a death rate of $5$ per $1000$ people.) The cost of the premium is $\$ 200$ per year, and the death benefit is $\$ 50,000$ What is the expected profit or loss for the insurance company?

The table shows the numbers of accidents, fatalities, hours flown, and miles flown for U.S. airlines with scheduled service in selected years.

$$\begin{array}{c|c|c|c|c} \text { Year } & \text { Accidents } & \text { Fatalities } & \begin{array}{c} \text { Hours flown } \\ \text { (millions) } \end{array} & \begin{array}{c} \text { Miles flown } \\ \text { (billions) } \end{array} \\ \hline 2000 & 49 & 89 & 16.7 & 7.1 \\ 2005 & 34 & 22 & 18.7 & 7.8 \\ 2010 & 28 & 0 & 17.2 & 7.3 \\ 2015 & 27 & 0 & 17.4 & 7.6 \end{array}$$

a. Compute the accident rate per million hours flown in $2000$ and $2015$. By this measure, has travel on U.S. airlines become safer? b. Compute the accident rate per billion miles flown in $2000$ and $2015$. By this measure, has travel on U.S. airlines become safer?

The below table gives prizes and probabilities of winning (on a single $\$ 1$ ticket) for the Multi-State Powerball lottery.

Find out the expected value of the winnings for a single lottery ticket if the jackpot is $\$ 45$ million. How much can you expect to win or lose each year if you buy $20$ lottery tickets per week? Should you actually expect to win or lose this amount?

Use these data to help answer the following question:

• U.S. data ($2016$ estimates); car crash fatalities $=40,200$; car crash injuries $=4.6$ million; cost to society $=$ $\$ 430$ billion. (Source: National Safety Council.)
• Global data (estimated): car crash fatalities $\approx 1.3$ million; car crash injuries $\approx 50$ million; car crash fatalities with victims under age $25=400,000$. (Source: Association for Safe International Road Travel.)

Globally, how many young people (under age $25$) die in car crashes each day on average?

Use these data to help answer the following question:

• U.S. data ($2016$ estimates); car crash fatalities $=40,200$; car crash injuries $=4.6$ million; cost to society $=$ $\$ 430$ billion. (Source: National Safety Council.)
• Global data (estimated): car crash fatalities $\approx 1.3$ million; car crash injuries $\approx 50$ million; car crash fatalities with victims under age $25=400,000$. (Source: Association for Safe International Road Travel.)

About how often (once every $x$ minutes or $x$ seconds) on average is a person killed in a car crash in the United States? How often is a person injured in a car crash?

The table below gives recent prizes and probabilities of winning (on a single $\$ 1$ ticket) for the Multi-State Powerball lottery. Find the expected value of the winnings for a single lottery ticket, given the following grand prizes. Then state how much you can expect to win or lose if you spend $\$ 365$ per year on the lottery. Briefly discuss your results.

The jackpot is $\$ 30$ million.

The table below gives recent prizes and probabilities of winning (on a single $\$ 1$ ticket) for the Multi-State Powerball lottery. Find the expected value of the winnings for a single lottery ticket, given the following grand prizes. Then state how much you can expect to win or lose if you spend $\$ 365$ per year on the lottery. Briefly discuss your results.

The jackpot is $\$ 10$ million.

In a large casino, the house wins on its blackjack tables with a probability of $50.7 \%$. All bets at blackjack are $1$ to $1$: If you win, you gain the amount you bet; if you lose, you lose the amount you bet.

a. If you bet $\$ 1$ on each hand what is the expected value to you of a single game? What is the house edge?

b. If you played $100$ games of blackjack in an evening, betting $\$ 1$ on each hand, how much should you expect to win or lose? Explain.

c. If you played $100$ games of blackjack in an evening, betting $\$ 5$ on each hand, how much should you expect to win or lose? Explain.

d. If patrons bet $\$ 1,000,000$ on blackjack in one evening, how much should the casino expect to earn? Explain.

Suppose you toss a fair coin $100$ times, getting $42$ heads and $58$ tails, which is $16$ more tails than heads. a. Explain why, on your next toss, the difference between the numbers of heads and tails is as likely to grow to $17$ as it is to shrink to $15$. b. Extend your explanation from part (a) to explain why, if you toss the coin $1000$ more times, the final difference between the numbers of heads and tails is as likely to be larger than $16$ as it is to be smaller than $16$. c. Suppose you are betting on heads with each coin toss. After the first $100$ tosses, you are well on the losing side. Explain why, if you continue to bet, you will most likely remain on the losing side. How is this answer related to the gambler's fallacy?

Find out the expected value (to the company) per policy sold. If the company sells $10,000$ policies, what is the expected profit or loss? Explain. An insurance policy sells for $\$ 600$. Based on past data, an average of $1$ in $50$ policyholders will file a $\$ 5000$ claim, an average of $1$ in $100$ policyholders will file a $\$ 10,000$ claim, and an average of $1$ in $200$ policyholders will file a $\$ 30,000$ claim.

Find out the expected value (to you) of the described game. Would you expect to win or lose money in a $1$ game? In $100$ games? Explain. You are given $5$ to $1$ odds against tossing three heads with three coins, meaning you win $\$ 5$ if you succeed and you lose $\$ 1$ if you fail.

Find the expected value (to you) of the described game. Would you expect to win or lose money in $1$ game? In $100$ games? Explain.

You are given $9$ to $1$ odds against tossing three heads with three coins, meaning you win $\$ 9$ if you succeed and you lose $\$ 1$ if you fail.

You are given $9$ to $1$ odds against tossing three heads with three coins, meaning you win $\$ 9$ if you succeed and you lose $\$ 1$ if you fail.

Decide whether the following statement is true or false. Explain your reasoning. I've lost $\$ 750$ so far today on roulette. I'm going to play a little longer so that $1$ can reduce my losses to $\$ 500$.