Chance. May 11, Chance Behavior The Idea of Probability Myths About Chance Behavior The Real Law of Averages Personal Probabilities

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1 Chance May 11, 2012 Chance Behavior The Idea of Probability Myths About Chance Behavior The Real Law of Averages Personal Probabilities 1.0 Chance Behavior 16 pre-verbal infants separately watch a puppet show. 14 out of 16 chose the helper toy. If infants have no genuine preference for either toy, how often would we see at least 14 out of 16 choosing the helper? helper hinderer

2 1.0 Chance Behavior Chance (or random) events exhibit chance behavior. Chance behavior is unpredictable in the short run, but has a regular predictable pattern in the long run. Probability helps us to describe chance behavior: it is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions. Long-run relative frequency 2.0 The Idea of Probability Toss a coin or choose a simple random sample. The result cannot be predicted in advance. But there is still a regular pattern in the results. The following figure shows the results of tossing a coin 1,000 times: HTTHHHT...

3 2.1 What Probability Does Not Say The probability of a head in tossing a fair coin is 0.5. This means that as we make more tosses, the proportion of heads will eventually get close to 0.5. This does not mean that the count of heads will get close to half the number of tosses. Why? Number of Proportion of Number of of Tosses of heads heads , , , ,704 The myth of short-run regularity. Our intuition tells us that randomness should also be regular in the short run. When it isn t, we look for some explanation other than chance variation. hot hand in basketball Toss a (fair) coin 10 times and record heads (H) or tails (T) on each toss HTHTTHHTHT TTTTTHHHHH HHHHHHHHHH Which one of these outcome is most probable? Least probable?

4 The myth of surprising coincidences. When something unusual happens, we say wasn t this particular outcome unlikely? On Nov. 18, 2006 O.S.U. beat Michigan in football by a score of 42 to 39. That same day, the winning numbers in the Pick 4 Ohio lottery were 4239! This is very unlikely. What is not so unlikely is that sometime during the 2006 season the winning number of some state lottery would match the recent score of some professional, college, or high school football game involving a team in the state. The myth of the law of averages. If an outcome has not happened on several previous opportunities, it is due to happen. Gambler s fallacy The occurrence of one extreme will be balanced by that of the other extreme so as to maintain the normal average. Dear Abby: My husband and I just had our eighth child. Another girl, and I am one very disappointed woman. I suppose I should thank God she was healthy, but Abby, this one was supposed to have been a boy. Even the doctor told me that the law of averages was in our favor 100 to one.

5 4.0 The Real Law of Averages Definition The law of averages a.k.a. law of large numbers states that in a large number of independent repetitions of a random phenomenon (such as coin tossing), averages or proportions are likely to become more stable as the number of trials increases, whereas sums or counts are likely to become more variable. The law of averages tells us that if we take a large S.R.S. from the population, the sample proportion p-hat will be very stable and close to the population parameter p. 5.0 Personal Probabilities Definition A personal probability of an outcome is a number between 0 and 1 that expresses an individual s judgment of how likely an outcome is. Personal probabilities are not limited to repeatable settings. They are useful because we base decisions on them. Because they express individual opinions, they aren t right or wrong. Why call personal opinions probabilities? They obey some of the same mathematical rules. Often, (not always), they are based on opinions from repeated observations.