6. Playing Pick 4. The Pick 4 games in many state lotteries announce a four-digit winning number each day. The winning number is essentially a four-digit group -om a table of random digits. You win if your choice matches the winning digits, in exact order. The winnings are divided among all players who matched the winning digits. That suggests a way to get an edge. (a) The winning number might be, for example, either 2873 or 9999. Explain why these two outcomes have exactly the same probability. (It is 1 in 10,000.) (b) If you asked many people which outcome is more likely to be the randomly chosen winning number, most would favor one of them. Use the information in this chapter to say which one and to explain why. If you choose a number that people think is unlikely, you have the same chance to win, but you will win a larger amount because few other people will choose your number. 7. An eerie coincidence? An October 6, 2002, ABC News article reported that the winning New York State lottery numbers on the one-year anniversary of the attacks on America were 911. Should this fact surprise you? Explain your answer. 8. In the long run. Probability works not by compensating for imbalances but by overwhelming them. Suppose that the rst 10 tosses of a coin give 10 tails and that tosses after that are exactly half heads and half tails. (Exact balance is unlikely, but the example illustrates how the rst 10 outcomes are swamped by later outcomes.) What is the proportion of heads after the rst 10 tosses? What is the proportion of heads aer 100 tosses if half of the last 90 produce heads (45 heads)? What is the proportion of heads aer 1000 tosses if half of the last 990 produce heads? What is the proportion of heads after 10,000 tosses if half of the last 9990 produce heads