Problem 3
Pick a card, any card. You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win \(\$ 5\). For any club, you win \(\$ 10\) plus an extra \(\$ 20\) for the ace of clubs. a) Create a probability model for the amount you win. b) Find the expected amount you'll win. c) What would you be willing to pay to play this game?
Problem 4
You bet! You roll a die. If it comes up a 6 , you win \(\$ 100\). If not, you get to roll again. If you get a 6 the second time, you win \(\$ 50\). If not, you lose. a) Create a probability model for the amount you win. b) Find the expected amount you'll win. c) What would you be willing to pay to play this game?
Problem 5
Kids. A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. (Assume boys and girls are equally likely.) a) Create a probability model for the number of children they might have. b) Find the expected number of children. c) Find the expected number of boys they'll have.
Problem 6
Carnival. A carnival game offers a \(\$ 100\) cash prize for anyone who can break a balloon by throwing a dart at it. It costs \(\$ 5\) to play, and you're willing to spend up to \(\$ 20\) trying to win. You estimate that you have about a \(10 \%\) chance of hitting the balloon on any throw. a) Create a probability model for this carnival game. b) Find the expected number of darts you'll throw. c) Find your expected winnings.
Problem 7
Software. A small software company bids on two contracts. It anticipates a profit of \(\$ 50,000\) if it gets the larger contract and a profit of \(\$ 20,000\) on the smaller contract. The company estimates there's a \(30 \%\) chance it will get the larger contract and a \(60 \%\) chance it will get the smaller contract. Assuming the contracts will be awarded independently, what's the expected profit?
Problem 8
Racehorse. A man buys a racehorse for \(\$ 20,000\) and enters it in two races. He plans to sell the horse afterward, hoping to make a profit. If the horse wins both races, its value will jump to \(\$ 100,000\). If it wins one of the races, it will be worth \(\$ 50,000\). If it loses both races, it will be worth only \(\$ 10,000\). The man believes there's a \(20 \%\) chance that the horse will win the first race and a \(30 \%\) chance it will win the second one. Assuming that the two races are independent events, find the man's expected profit.
Problem 17
Defects. A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips, etc.). While none had more than three of these defects, \(7 \%\) had three, \(11 \%\) two, and \(21 \%\) one defect. Find the expected number of appearance defects in a new car and the standard deviation.
Problem 18
Insurance. An insurance policy costs \(\$ 100\) and will pay policyholders \(\$ 10,000\) if they suffer a major injury (resulting in hospitalization) or \(\$ 3000\) if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2000 policyholders may have a major injury, and 1 in 500 a minor injury only. a) Create a probability model for the profit on a policy. b) What's the company's expected profit on this policy? c) What's the standard deviation?
Problem 19
Cancelled flights. Mary is deciding whether to book the cheaper flight home from college after her final exams, but she's unsure when her last exam will be. She thinks there is only a \(20 \%\) chance that the exam will be scheduled after the last day she can get a seat on the cheaper flight. If it is and she has to cancel the flight, she will lose \(\$ 150\). If she can take the cheaper flight, she will save \(\$ 100\). a) If she books the cheaper flight, what can she expect to gain, on average? b) What is the standard deviation?
Problem 20
Day trading. An option to buy a stock is priced at \(\$ 200\). If the stock closes above 30 on May 15 , the option will be worth \(\$ 1000\). If it closes below 20 , the option will be worth nothing, and if it closes between 20 and 30 (inclusively), the option will be worth \(\$ 200\), A trader thinks there is a \(50 \%\) chance that the stock will close in the \(20-30\) range, a \(20 \%\) chance that it will close above 30 , and a \(30 \%\) chance that it will fall below 20 on May 15 .a) Should she buy the stock option? b) How much does she expect to gain? c) What is the standard deviation of her gain?
