91Ó°ÊÓ

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?

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.

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?

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?

Farmers' market. A farmer has \(100 \mathrm{lb}\) of apples and \(50 \mathrm{lb}\) of potatoes for sale. The market price for apples (per pound) each day is a random variable with a mean of \(0.5\) dollars and a standard deviation of \(0.2\) dollars. Similarly, for a pound of potatoes, the mean price is \(0.3\) dollars and the standard deviation is \(0.1\) dollars. It also costs him 2 dollars to bring all the apples and potatoes to the market. The market is busy with eager shoppers, so we can assume that he'll be able to sell all of each type of produce at that day's price. a) Define your random variables, and use them to express the farmer's net income. b) Find the mean. c) Find the standard deviation of the net income. d) Do you need to make any assumptions in calculating the mean? How about the standard deviation?

Weightlifting. The Atlas BodyBuilding Company (ABC) sells "starter sets" of barbells that consist of one bar, two 20-pound weights, and four 5 -pound weights. The bars weigh an average of 10 pounds with a standard deviation of \(0.25\) pounds. The weights average the specified amounts, but the standard deviations are \(0.2\) pounds for the 20-pounders and \(0.1\) pounds for the 5 -pounders. We can assume that all the weights are normally distributed. a) \(\mathrm{ABC}\) ships these starter sets to customers in two boxes: The bar goes in one box and the six weights go in another. What's the probability that the total weight in that second box exceeds \(60.5\) pounds? Define your variables clearly and state any assumptions you make. b) It costs \(\mathrm{ABC} \$ 0.40\) per pound to ship the box containing the weights. Because it's an odd-shaped package, though, shipping the bar costs \(\$ 0.50\) a pound plus a \(\$ 6.00\) surcharge. Find the mean and standard deviation of the company's total cost for shipping a starter set. c) Suppose a customer puts a 20-pound weight at one end of the bar and the four 5 -pound weights at the other end. Although he expects the two ends to weigh the same, they might differ slightly. What's the probability the difference is more than a quarter of a pound?