17.8 Nickels spinning. Hold a nickel upright on its edge under your forefinger on a hard surface, then snap it with your other forefinger so that it spins for some time before falling. Based on 50 spins, estimate the probability of heads.

17.9 Nickels falling over. You may feel that it is obvious that the probability of a head in tossing a coin is about 1-in-2 because the coin has two faces. Such opinions are not always correct. The previous exercise asked you to spin a nickel rather than toss it—that changes the probability of a head. Now try another variation. Stand a nickel on edge on a hard, flat surface. Pound the surface with your hand so that the nickel falls over. What is the probability that it falls with heads upward? Make at least 50 trials to estimate the probability of a head.

17.10 Random digits. The table of random digits (Table A) was produced by a random mechanism that gives each digit probability 0.1 of being a 0. What proportion of the 400 digits in lines 120 to 129 in the table are 0s? This proportion is an estimate, based on 400 repetitions, of the true probability, which in this case is known to be 0.1.

17.11 How many tosses to get a head? When we toss a penny, experience shows that the probability (long-term proportion) of a head is close to 1-in-2. Suppose now that we toss the penny repeatedly until we get a head. What is the probability that the first head comes up in an odd number of tosses (one, three, five, and so on)? To find out, repeat this experiment 50 times, and keep a record of the number of tosses needed to get a head on each of your 50 trials.

17.12 Tossing a thumbtack. Toss a thumbtack on a hard surface 50 times. How many times did it land with the point up? What is the approximate probability of landing point up?

17.13 Rolling dice. Roll a pair of dice 100 times. How many times did you roll a 5? What is the approximate probability of rolling a 5?

17.14 Straight. You read in a book on poker that the probability of being dealt a straight (excluding a straight flush or royal flush) in a five-card poker hand is about 0.00393. Explain in simple language what this means.

17.15 From words to probabilities. Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. (The probability is usually a more exact measure of likelihood than is the verbal statement.)

0 0.01  0.3 0.6 0.99 1

17.16 Winning a baseball game. Over the period from 1965 to 2014, the champions of baseball’s two major leagues won 63% of their home games during the regular season. At the end of each season, the two league champions meet in the baseball World Series. Would you use the results from the regular season to assign probability 0.63 to the event that the home team wins a World Series game? Explain your answer.

17.17 Will you have an accident? The probability that a randomly chosen driver will be involved in an accident in the next year is about 0.2. This is based on the proportion of millions of drivers who have accidents. “Accident” includes things like crumpling a fender in your own driveway, not just highway accidents.

17.18 Marital status. Based on 2010 census data, the probability that a randomly chosen woman over 64 years of age is divorced is about 0.11. This probability is a long-run proportion based on all the millions of women over 64. Let’s suppose that the proportion stays at 0.11 for the next 45 years. Bridget is now 20 years old and is not married.

17.19 Personal probability versus data. Give an example in which you would rely on a probability found as a long-term proportion from data on many trials. Give an example in which you would rely on your own personal probability.

17.20 Personal probability? When there are few data, we often fall back on personal probability. There had been just 24 space shuttle launches, all successful, before the Challenger disaster in January 1986. The shuttle program management thought the chances of such a failure were only 1 in 100,000.

17.21 Personal random numbers? Ask several of your friends (at least 10 people) to choose a four-digit number “at random.” How many of the numbers chosen start with 1 or 2? How many start with 8 or 9? (There is strong evidence that people in general tend to choose numbers starting with low digits.)

17.22 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 from 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.

17.23 Surprising? During the NBA championship series in 2015, news media reported that LeBron James and Stephen Curry, both winners of the NBA most valuable player award, were born in the same hospital in Akron, Ohio. That a pair of winners of the NBA most valuable player award were also born in the same hospital was reported as an extraordinarily improbable event. Should this fact (that both were winners of the most valuable player award and both were born in the same hospital) surprise you? Explain your answer.

17.24 An eerie coincidence? An October 6, 2002, an 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.

17.25 Curry’s free throws. The basketball player Stephen Curry is the all-time career free-throw shooter among active players. He makes 90.0% of his free throws. In today’s game, Curry misses his first two free throws. The TV commentator says, “Curry’s technique looks out of rhythm today.” Explain why the claim that Curry’s technique has deteriorated is not justified.

17.26 In the long run. Probability works not by compensating for imbalances but by overwhelming them. Suppose that the first 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 first 10 outcomes are swamped by later outcomes.) What is the proportion of heads after the first 10 tosses? What is the proportion of heads after 100 tosses if half of the last 90 produce heads (45 heads)? What is the proportion of heads after 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?

17.27 The “law of averages.“ The baseball player Miguel Cabrera gets a hit about 32.1% of the time over an entire season. After he has failed to hit safely in nine straight at-bats, the TV commentator says, “Miguel is due for a hit by the law of averages.” Is that right? Why?

17.28 Snow coming. A meteorologist, predicting below-average snowfall this winter, says, “First, in looking at the past few winters, there has been above-average snowfall. Even though we are not supposed to use the law of averages, we are due.” Do you think that “due by the law of averages” makes sense in talking about the weather? Explain.

17.29 An unenlightened gambler.

17.30 Reacting to risks. The probability of dying if you play high school football is about 10 per million each year you play. The risk of getting cancer from asbestos if you attend a school in which asbestos is present for 10 years is about 5 per million. If we ban asbestos from schools, should we also ban high school football? Briefly explain your position.

17.31 Reacting to risks. National newspapers such as USA Today and the New York Times carry many more stories about deaths from airplane crashes than about deaths from motor vehicle crashes. Motor vehicle accidents killed about 32,700 people in the United States in 2013. Crashes of all scheduled air carriers worldwide, including commuter carriers, killed 266 people in 2013, and only one of these involved a U.S. air carrier.

17.32 What probability doesn’t say. The probability of a head in tossing a coin is 1-in-2. This means that as we make more tosses, the proportion of heads will eventually get close to 0.5. It does not mean that the count of heads will get close to one-half the number of tosses. To see why, imagine that the proportion of heads is 0.49 in 100 tosses, 0.493 in 1000 tosses, 0.4969 in 10,000 tosses, and 0.49926 in 100,000 tosses of a coin. How many heads came up in each set of tosses? How close is the number of heads to half the number of tosses?