Chapter 17: Thinking About Chance Thought Question Flip a coin 20 times and record the results of each flip (H or T). Do we know what will come up before we flip the coin? If we flip a coin many, many times, what sort of pattern do we expect to see in the results? 1
Probability Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. Example: Rolling a Die Roll a die. If it is unbiased, then each side (1, 2, 3, 4, 5, 6) should occur with the same chance. After many, many rolls, what do you think the proportion of 2 s which occur is close to? 1/6 The following graphs show the proportion of tosses which result in a 2 ; each plot represents 1, 000 rolls of a fair die. 2
Terminology Random phenomenon: individual outcomes are uncertain but there is still a regular distribution of outcomes in a large number of repetitions. Probability of an outcome: a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions. Properties of Probability The higher the probability for a particular outcome (i.e., the closer it is to 1), the more likely it is to occur. The lower the probability for a particular outcome (i.e., the closer it is to 0), the less likely it is to occur. Outcomes with probability 1 will always occur. Outcomes with probability 0 will never occur. 3
Myths About Probability: Short Term Regularity Consider the following situations: You toss a coin 10 times and record the results. Here are some possible outcomes: HTHTTHHTHT TTTTTHHHHH HHHHHHHHHH You say that HTHTTHHTHT is the most likely outcome and TTTTTHHHHH and HH- HHHHHHHH don t seem random and are less likely. What is incorrect about this reasoning? All three outcomes are equally probable - each coin flip is independent from any previous coin flip. At a roulette wheel, the following colors have been observed in the last 15 plays: R G B R B R B R B B B B B B B We walk to the table with this information and bet on black because there has been a streak of blacks. What is incorrect about this reasoning? Each color is still equally probable. The roulette wheel isn t influenced by past colors. 4
Myths About Probability: Surprising Coincidence Consider the following situations: In 1986, Evelyn Marie Adams won the NJ state lottery for a second time (1.5 and 3.9 million dollar payoffs). Robert Humphries (PA) won his second lottery two years later (6.8 million total). How likely are you to win the lottery twice? Very unlikely! Is it surprising that someone won the lottery twice? No. How likely do you think it is that at least two people sitting in this room have the same birthday? Not very - but in fact there s about a 40% chance for 20 people. 5
Myths About Probability: Law of Averages Consider the following situations: A baseball team is behind in the 9th inning. To rally the crowd, the announcer says, At bat is Sandburg, who is hitting.250 for the season. Today he s 0 for 3, so he is due for hit. What is incorrect about this reasoning? Each hit happens independently of previous hits. He has the same chance of making a hit regardless of his previous at-bats. Empirical data suggests that the boys and girls are born at roughly the same rate (about 50/50). A couple s first three children are boys. They assume their next child is very likely to be a girl. What is incorrect about this reasoning? The child is equally likely to be a boy or a girl. It doesn t matter what gender the older children are. 6
Truths About Probability: Law of Averages Law of Averages (Law of Large Numbers) As the number of repetitions increases (i.e., in the long run), the sample proportion ˆp of successes approaches the true probability of success p. Example: Poll A Fox News poll, taken on June 29, 2006, reported the results from an SRS of n = 900 adults nationwide. Interviewers asked the following question: Do you approve or disapprove of the way George W. Bush is handling his job as president? Of the 900 adults in the sample, 369 responded by stating they approve of the President s handling of his job. Calculate the margin of error for this sample size of 900 people. margin of error 1 n = 1 900 0.033. Calculate the margin of error for a sample size of 2,000 people. margin of error 1 n = 1 2000 0.022. Calculate the margin of error for a sample size of 5,000 people. margin of error 1 n = 1 5000 0.014. Calculate the margin of error for a sample size of 14,000 people. margin of error 1 n = 1 14000 0.008. As the sample size increases, the margin of error decreases, and so the sample proportion gets closer to the true proportion 7
Personal Probabilities What if a situation doesn t repeat many times? Terminology Personal probability of an outcome: a number between 0 and 1 that expresses an individual s judgment of how likely an outcome is. Example: Grade At this point in the class, what do you think your chance is of earning a B or higher in this class? Express your answer as a probability. Comparison of Probabilities Personal probability: scientific). subjective; based on personal opinion (and, hence, is often not Long term regularity: based on the notion of repeated trials; e.g., what happens in the long term? There are methods called Bayes s procedures for using data to update personal probabilities. 8
Probability and Risk Why don t the public and experts agree on what is risky and what isn t? Example: Abductions vs. Car Accidents The chance of a child being killed by a non-family member is (approximately) between 1 in 364,000 and 1 in 1 million. A child s risk of dying in a car accident is twenty-five to seventy-five times greater. Do you think parents worry more about their child being abducted and murdered by a stranger, or about their child dying in a car accident? Why? Parents worry more about abduction. Maybe this is because it feels more out of their control. Reasons Why We Don t Think Clearly About Risk We feel safer when a risk seems under our control than when we cannot control it. It is hard to comprehend very small probabilities. It is very hard to estimate certain risks. Some people may feel that experts underestimate risks. 9
Chapter 17 Exercises 1. You read in a book on poker that the probability of being dealt three of kind in a five-card poker hand is about 1/47. Explain in simple terms what this means. If you dealt many five-card poker hands, you d expect about 1 out of every 47 to have three of a kind. 2. 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 44,000 people in the United States in 2007. Crashes of all scheduled air carriers worldwide, including commuter carriers, killed 587 people in 2007, and only 1 of these involved a U.S. air carrier. (a) Why do the news media give more attention to airplane crashes? They happen less frequently and are more dramatic, as usually everyone on the plane (dozens or hundred of people) die. (b) How does news coverage help explain why many people consider flying more dangerous than driving? People see more stories about airplane crashes on the news, so they are more likely to be afraid of dying in an airplane crash rather than dying in a car crash. 10