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Random outcomes For each of the following scenarios, decide if the outcome is random. a) Flip a coin to decide who takes out the trash. Is who takes out the trash random? b) A friend asks you to quickly name a professional sports team. Is the sports team named random? c) Names are selected out of a hat to decide roommates in a dormitory. Is your roommate for the year random?

Geography An elementary school teacher with 25 students plans to have each of them make a poster about two different states. The teacher first numbers the states (in alphabetical order, from 01-Alabama to 50 -Wyoming), then uses a random number table to decide which states each kid gets. Here are the random digits: $$45921017102289237076$$ a) Which two state numbers does the first student get? b) Which two state numbers go to the second student?

Play the Iottery Some people play state-run lotteries by always playing the same favorite "lucky" number. Assuming that the lottery is truly random, is this strategy better, worse, or the same as choosing different numbers for each play? Explain.

Multiple choice You take a quiz with 6 multiple choice questions. After you studicd, you estimated that you would have about an \(80 \%\) chance of getting any individual question right. What are your chances of getting them all right? Use at least 20 trials.

Beat the lottery Many states run lotteries to raise money. A Web site advertises that it knows "how to increase YOUR chances of Winning the Lottery." They offer several systems and criticize others as foolish. One system is called Lucky Numbers. People who play the Lucky Numbers system just pick a "lucky" number to play, but maybe some numbers are luckier than others. Let's use a simulation to see how well this system works. To make the situation manageable, simulate a simple lottery in which a single digit from 0 to 9 is selected as the winning number. Pick a single value to bet, such as 1 and keep playing it over and over. You'll want to run at least 100 trials. (If you can program the simulations on a computer, run several hundred. Or generalize the questions to a lottery that chooses two- or three-digit numbers - for which you'll need thousands of trials.) a) What proportion of the time do you expect to win? b) Would you expect better results if you picked a "luckier" number, such as \(7 ?\) (Try it if you don't know.) Explain.

The family Many couples want to have both a boy and a girl. If they decide to continue to have children until they have one child of each sex, what would the average family size be? Assume that boys and girls are equally likely.

Parcheesi You are three spaces from a win in Parcheesi. On each turn, you will roll two dice. To win, you must roll a total of 3 or roll a 3 on one of the dice. How many turns might you expect this to take?