Problem 1
For each of the following, list the sample space and tell whether you think the events are equally likely: a) Toss 2 coins; record the order of heads and tails. b) A family has 3 children; record the number of boys. c) Flip a coin until you get a head or 3 consecutive tails; record each flip. d) Roll two dice; record the larger number.
Problem 2
For each of the following, list the sample space and tell whether you think the events are equally likely: a) Roll two dice; record the sum of the numbers. b) A family has 3 children; record each child's sex in order of birth. c) Toss four coins; record the number of tails. d) Toss a coin 10 times; record the length of the longest run of heads.
Problem 3
A casino claims that its roulette wheel is truly random. What should that claim mean?
Problem 4
The weather reporter on TV makes predictions such as a \(25 \%\) chance of rain. What do you think is the meaning of such a phrase?
Problem 6
After an unusually dry autumn, a radio announcer is heard to say, "Watch out! We'll pay for these sunny days later on this winter." Explain what he's trying to say, and comment on the validity of his reasoning.
Problem 7
A batter who had failed to get a hit in seven consecutive times at bat then hits a game-winning home run. When talking to reporters afterward, he says he was very confident that last time at bat because he knew he was "due for a hit." Comment on his reasoning.
Problem 8
Commercial airplanes have an excellent safety record. Nevertheless, there are crashes occasionally, with the loss of many lives. In the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are afraid to risk flying. a) A travel agent suggests that since the law of averages makes it highly unlikely to have two plane crashes within a few weeks of each other, flying soon after a crash is the safest time. What do you think? b) If the airline industry proudly announces that it has set a new record for the longest period of safe flights, would you be reluctant to fly? Are the airlines due to have a crash?
Problem 9
Insurance companies collect annual payments from homeowners in exchange for paying to rebuild houses that burn down. a) Why should you be reluctant to accept a \(\$ 300\) payment from your neighbor to replace his house should it burn down during the coming year? b) Why can the insurance company make that offer?
Problem 10
On January 20,2000 , the International Gaming Technology company issued a press release: (LAS VEGAS, Nev.) - Cynthia Jay was smiling ear to ear as she walked into the news conference at The Desert Inn Resort in Las Vegas today, and well she should. Last night, the 37 -year-old cocktail waitress won the world's largest slot jackpot - \(\$ 34,959,458\) - on a Megabucks machine. She said she had played \(\$ 27\) in the machine when the jackpot hit. Nevada Megabucks has produced 49 major winners in its 14-year history. The top jackpot builds from a base amount of \(\$ 7\) million and can be won with a \(3-\operatorname{coin}(\$ 3)\) bet. a) How can the Desert Inn afford to give away millions of dollars on a \(\$ 3\) bet? b) Why did the company issue a press release? Wouldn't most businesses want to keep such a huge loss quiet?
Problem 13
Suppose that \(46 \%\) of families living in a certain county own a car and \(18 \%\) own an SUV. The Addition Rule might suggest, then, that \(64 \%\) of families own either a car or an SUV. What's wrong with that reasoning?
