The natural next question: When does our rule-of-thumb approach to calculating the odds lead to serious miscalculation? One answer was: Whenever people are asked to evaluate anything with a random component to it. It wasn’t enough that the uncertain event being judged resembled the parent population, wrote Danny and Amos. “The event should also reflect the properties of the uncertain process by which it is generated.” That is, if a process is random, its outcome should appear random. They didn’t explain how people’s mental model of “randomness” was formed in the first place. Instead they said, Let’s look at judgments that involve randomness, because we psychologists can all pretty much agree on people’s mental model of it.
Londoners in the Second World War thought that German bombs were targeted, because some parts of the city were hit repeatedly while others were not hit at all. (Statisticians later showed that the distribution was exactly what you would expect from random bombing.) People find it a remarkable coincidence when two students in the same classroom share a birthday, when in fact there is a better than even chance, in any group of twenty-three people, that two of its members will have been born on the same day. We have a kind of stereotype of “randomness” that differs from true randomness. Our stereotype of randomness lacks the clusters and patterns that occur in true random sequences. If you pass out twenty marbles randomly to five boys, they are actually more likely to each receive four marbles (column II), than they are to receive the combination in column I, and yet American college students insisted that the unequal distribution in column I was more likely than the equal one in column II. Why? Because column II “appears too lawful to be the result of a random process. . . . ”
A suggestion arose from Danny and Amos’s paper: If our minds can be misled by our false stereotype of something as measurable as randomness, how much might they be misled by other, vaguer stereotypes?
The average heights of adult males and females in the U.S. are, respectively, 5 ft. 10 in. and 5 ft. 4 in. Both distributions are approximately normal with a standard deviation of about 2.5 in.§
An investigator has selected one population by chance and has drawn from it a random sample.
What do you think the odds are that he has selected the male population if
1. The sample consists of a single person whose height is 5 ft. 10 in.?
2. The sample consists of 6 persons whose average height is 5 ft. 8 in.?
The odds most commonly assigned by their subjects were, in the first case, 8:1 in favor and, in the second case, 2.5:1 in favor. The correct odds were 16:1 in favor in the first case, and 29:1 in favor in the second case. The sample of six people gave you a lot more information than the sample of one person. And yet people believed, incorrectly, that if they picked a single person who was five foot ten, they were more likely to have picked from the population of men than had they picked six people with an average height of five foot eight. People didn’t just miscalculate the true odds of a situation: They treated the less likely proposition as if it were the more likely one. And they did this, Amos and Danny surmised, because they saw “5 ft. 10 in.” and thought: That’s the typical guy! The stereotype of the man blinded them to the likelihood that they were in the presence of a tall woman.
A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50 percent of all babies are boys. The exact percentage of baby boys, however, varies from day to day. Sometimes it may be higher than 50 percent, sometimes lower.
For a period of 1 year, each hospital recorded the days on which more than 60 percent of the babies born were boys. Which hospital do you think recorded more such days? Check one:
— The larger hospital
— The smaller hospital
— About the same (that is, within 5 percent of each other)
People got that one wrong, too. Their typical answer was “same.” The correct answer is “the smaller hospital.” The smaller the sample size, the more likely that it is unrepresentative of the wider population. “We surely do not mean to imply that man is incapable of appreciating the impact of sample size on sampling variance,” wrote Danny and Amos. “People can be taught the correct rule, perhaps even with little difficulty. The point remains that people do not follow the correct rule, when left to their own devices.”
To which a bewildered American college student might reply: All these strange questions! What do they have to do with my life? A great deal, Danny and Amos clearly believed. “In their daily lives,” they wrote, “people ask themselves and others questions such as: What are the chances that this 12-year-old boy will grow up to be a scientist? What is the probability that this candidate will be elected to office? What is the likelihood that this company will go out of business?” They confessed that they had confined their questions to situations in which the odds could be objectively calculated. But they felt fairly certain that people made the same mistakes when the odds were harder, or even impossible, to know. When, say, they guessed what a little boy would do for a living when he grew up, they thought in stereoypyes. If he matched their mental picture of a scientist, they guessed he’d be a scientist—and neglect the prior odds of any kid becoming a scientist.
Of course, you couldn’t prove that people misjudged the odds of a situation when the odds were extremely difficult or even impossible to know. How could you prove that people came to the wrong answer when a right answer didn’t exist? But if people’s judgments were distorted by representativeness when the odds were knowable, how likely was it that their judgments were any better when the odds were a total mystery?
* * *
Danny and Amos had their first big general idea—the mind had these mechanisms for making judgments and decisions that were usually useful but also capable of generating serious error. The next paper they produced inside the Oregon Research Institute described a second mechanism, an idea that had come to them just a couple of weeks after the first. “It wasn’t all representativeness,” said Danny. “There was something else going on. It wasn’t just similarity.” The new paper’s title was once again more mystifying than helpful: “Availability: A Heuristic for Judging Frequency and Probability.” Once again, the authors came with news of the results of questions that they had posed to students, mostly at the University of Oregon, where they now had an endless supply of lab rats. They’d gathered a lot more kids in classrooms and asked them, absent a dictionary or any text, to answer these bizarre questions:
The frequency of appearance of letters in the English language was studied. A typical text was selected, and the relative frequency with which various letters of the alphabet appeared in the first and third positions of the words was recorded. Words of less than three letters were excluded from the count.
You will be given several letters of the alphabet, and you will be asked to judge whether these letters appear more often in the first or in the third position, and to estimate the ratio of the frequency with which they appear in these positions. . . .
Consider the letter K
Is K more likely to appear in
____the first position?
____the third position?
(check one)
My estimate for the ratio of these two values is:________:1