Steps in Assigning Probabilities Some probabilities we have direct intuitions about - e.g., my guess is that the percentage of faculty who own Hondas or Toyotas is about 40%. [P(H or T/F = 0.4] Other probabilities we get by making crude calculations - e.g., I guess about half of the above Toyotas/Hondas are pretty new. [P(N/(H or T) and F = 0.5] I can now calculate that the probability that a faculty member owns a relatively new H or T is about 20%. [ P(N and (H or T)/F) = 0.2]

Analogous Combinations of Estimating/Calculating Figuring out how much paint to buy for a room: I estimate the height and other linear dimensions of the room and then calculate the area of the walls. Figuring out the length of a big ball of string: I estimate the diameter of the ball and the diameter of the string and then calculate the length of the string.

Revisiting the Confusion about Inverses Here we have an intuition about one conditional probability and then try to use an oversimplified method for calculating the inverse. The two probabilities will be equal only when the size of the classes under consideration are of comparable size. Misuse of the heuristic assumption that the classes are about the same size can seriously bias the results. Put more formally, P(A/B) is approximately equal to P(B/A) when P(A) is approximately equal to P(B). Otherwise they will not be close in numerical value.

Revisiting the Confusion about Inverses Here we have an intuition about one conditional probability and then try to use an oversimplified method for calculating the inverse. The two probabilities will be equal only when the size of the classes under consideration are of comparable size. The heuristic assumption that the classes are about the same size can seriously bias the results.

Examples where the heuristic works and where it fails It works in the case of the Unitarian students who tend to be both Vegetarian and Death- penalty-opponents, at least as described in class. It obviously doesn’t work in the Tooth-brushing/Axe-murders example and people are not tempted to use it there. The Dope-smoking/Hard-drugs example falls in between and in that case people do have confusions about the likely value of the inverse.

Confusions about Conjunction and Disjunction The disjunction of A and B depends on the total area occluded by the two circles. People tend to underestimate the probability of A or B. The conjunction of A and B depends on the area of overlap. People tend to overestimate the probability of A and B, thinking perhaps about the average size of the two circles instead of the area of their intersection.

How to Explain the Linda/Bank- Teller/Feminist Probability Assignments Plous, following Tversky and Kahneman, describes this as a case of confusion about the relationship of the probability of a conjunction, namely that Linda is a bank teller and a feminist - P(B&F) - to the probability of each of the conjuncts, namely the probability that Linda is a feminist - P(F) - and the probability that Linda is a bank teller - P(B).

Explanation Two of the Linda Phenomenon Another possibility is that people revise their probability assignments in the process of doing the questionnaire. Maybe what people do is to first set the probability that Linda is a bank teller rather low, but as they imagine in more detail the heterogeneity of the class of bank tellers, they essentially raise the probability that Linda is a bank teller (given that some bank tellers could be feminists) and they record that answer only on the entry that prompts them to think about feminist bank tellers. Imagination can affect our probability assignments.

Explanation Three: Confusion about the Inverse Again Let L be the class of women who match the description of Linda rather closely. Now the following assignments seem plausible and do not violate the conjunction rule: P(L/F) > P(L/F&T) > P(L/T) What people report in the experiment, however, is: P(F/L) > P(F&T/L) > P(T/L) which does violate the rule.

A More Complicated Case of Confusion About the Inverse (Plous, p.131) Visual evidence suggests that the probability that a lump is cancerous is only 1%. A lab test that correctly classifies 80% of cancerous tumors and 90% of benign tumors says the lump is cancerous. Combining these bits of evidence, what should we conclude about the probability that the lump is cancerous? Is it high or low?