It evens out in the end The “beat the lottery” website of Exercise 25 notes that in the long run we expect each value to turn up about the same number of times. That leads to their recommended strategy. First, watch the lottery for a while, recording the winners. Then bet the value that has turned up the least, because it will need to turn up more often to even things out. If there is more than one “rarest” value, just take the lowest one (since it doesn’t matter). Simulating the simplified lottery described in Exercise 25, play many games with this system. What proportion of the time do you win?
Exercise 25
Beat the lottery Many states run lotteries to raise money. A website 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.
Exercise 23
Multiple choice You take a quiz with 6 multiple choice questions. After you studied, 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.