How People Assign Probabilities

• We now turn to a variety of studies that deal with how people reason about and with probabilities.

• Some of these patterns are not very directly connected with decision making, but it is useful to look at the overall picture re probability.

• Although a formal account of probability theory was not proposed until the 17th C., the behavior of people - and animals - is either explicitly or implicitly influenced by probability considerations.

• Once again we will compare normative and descriptive accounts.

Confusion about the Inverse

More Examples

• A similar diagram would probably represent the relationship between:

– Going to college and being an atheist

– Loving dogs and being President of the USA

– Drinking tea and being British

– Being an Eskimo and carving soapstone seals

– Being rather quiet and being a librarian

– Running a fever and having small pox

– Brushing your teeth and being an axe murderer

• Yet in some cases (which ones?) people infer a high probability from the fact that the converse is high.

Confusion about Conjunction

• Look at the previous diagram again. Note that the area where the circles overlap can never be larger than the area of either circle.

• What this means is that the probability of A & B can never be greater than the probability of A alone or that of B alone.

• But people sometimes think that the probability of the conjunction should be sort of half way in between the probabilities of the two conjuncts!

An Example

• Suppose you believe that the typical pot smoker inhales, namely that the probability of inhaling, given that you’re a pot smoker, is high.

• Then you might mistakenly reason as follows:

– You found papers in little Susie’s room? Bet you a buck, she’s smoking pot - and I’ll bet you two bucks that, unlike Clinton, our little Susie’s inhaling!

• But the probability of P&I cannot be greater than the probability of P simpliciter.

Ignoring Base Rates

• Suppose you are standing at Sample Gates at midnight and hear the sound of hoof beats.

• They sound just like a horse to you, but your friend, who has worked in a zoo, says they sound just like a zebra.

• Which is more probable, a horse is coming down Kirkwood Avenue or there is a zebra loose?

• To guide your intuitions, here is a photo from the Herald-Times:

More on Base Rates

• It is important to consider base rates when physicians diagnose diseases, bird watchers identify birds, or anthropologists interpret fossils or artifacts.

• However, people sometimes focus too much on base rates and ignore the peculiar features of the individual case or specimen in hand. This is a form of stereotyping.

• Let us now turn to some of the classic experiments on the confusions or mistakes discussed so far.

A panel of psychologists have given personality tests to a sample consisting of 100 successful men: 30 engineers and 70 lawyers. What is the probability for each of the following dossiers that the person described is an engineer?

• Jack is 45, married, four kids. He is conservative, careful, ambitious. He shows no interest in politics and spends much of his free time on hobbies such as carpentry, sailing and math puzzles.

• Dick is 30, married but no children. A man of high ability and high motivation, he promises to quite successful. He is well liked by his colleagues.

• X is a member of the sample drawn at random.

Ignoring the Base Rate

• In the above experiment, most people thought the description of Dick was neutral between their stereotypical pictures of lawyers and engineers and thus assigned that probability as 50%.

• But in this case the right answer should be the same as that assigned to a random draw - 30%.

• The value assigned to Jack should be above 30% but probably not as high as the 70-80% values actually assigned.

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

• Please rank the following by their probability, using 1 for the most probable and 8 for the least probable description of Linda :

– Is a teacher in elementary school

– Works in a bookstore and takes yoga classes

– Is active in the feminist movement

– Is a psychiatric social worker

– Is a member of the League of Women Voters

– Is a bank teller

– Is an insurance salesperson

– Is a bank teller and active in the feminist movement.

The Average Rankings Violate the Conjunction Law

(5.2) Is a teacher in elementary school

(3.3) Works in a bookstore and takes yoga classes

(2.1) Is active in the feminist movement

(3.1) Is a psychiatric social worker

(5.4) Is a member of the League of Women Voters

(6.2) Is a bank teller

(6.4) Is an insurance salesperson

(4.1) Is a bank teller and active in the feminist movement.

Forgetting about Regression to the Mean

• It takes a lot of luck (not talent) to throw three double sixes in a row. Just because Jones has done it once, we should not expect Jones to do it again.

• It takes considerable luck (as well as lots of talent) to sink ten three point shots in a game. If Smith does it in one game and then hits only seven the following week, we should not conclude that Smith was slacking off or being overconfident or whatever. Perhaps Smith’s production of seven is just around her average. Her talent is not decreasing - she was just lucky the week before.

The Sports Illustrated Jinx

• The above phenomenon is probably the explanation of the so-called “sophomore slump” in professional sports. Rookies of the year typically do less well their second year in the league.

• Part of the explanation may be that the opponents are now keying on them, but it’s probably also true that it takes luck (improbably fortuitous circumstances) to become rookie-of-the-year.

Applications to Learning and Prediction

• Some of the improvements in our performance are due to luck, not to an improvement in underlying skill. This can lead to confusion and the adoption of superstitious habits.

• It can also cause us to make seriously biased forecasts about future performance.

• Here are examples of each.

The Grumpy Flight Instructor when Told Praise is a Better Motivator than Blame

• “With all due respect, Sir, what you are saying is literally for the birds. I’ve often praised people warmly for beautifully executed maneuvers, and the next time they almost always do worse. And I’ve screamed at airmen for badly executed maneuvers, and by and large, the next time they improve. Don’t tell me that reward works and punishment doesn’t. My experience contradicts it.”

Suppose that scores on a high school academic achievement test are moderately related to college grade point averages. Given the percentiles below, what college GPA would you predict for a student who scored 725 on the high school achievement test?

Percentile Test Score GPA