The following experiment has been slightly modified for ease of blogging. You are given the following written description, which is assumed true:

Bill is 34 years old. He is intelligent, but unimaginative, compulsive, and generally lifeless. In school, he was strong in mathematics but weak in social studies and humanities.

No complaints about the description, please, this experiment was done in 1974. Anyway, we are interested in the probability of the following propositions, which may or may not be true, and are *not* mutually exclusive or exhaustive:

A: Bill is an accountant.B: Bill is a physician who plays poker for a hobby.C: Bill plays jazz for a hobby.D: Bill is an architect.E: Bill is an accountant who plays jazz for a hobby.F: Bill climbs mountains for a hobby.

Take a moment before continuing to *rank* these six propositions by probability, starting with the *most probable* propositions and ending with the *least probable* propositions. Again, the starting description of Bill is assumed true, but the six propositions may be true or untrue (they are not additional evidence) and they are not assumed mutually exclusive or exhaustive.

In a very similar experiment conducted by Tversky and Kahneman (1982), 92% of 94 undergraduates at the University of British Columbia gave an ordering with **A** > **E** > **C**. That is, the vast majority of subjects indicated that Bill was more likely to be an accountant than an accountant who played jazz, and more likely to be an accountant who played jazz than a jazz player. The ranking **E** > **C** was also displayed by 83% of 32 grad students in the decision science program of Stanford Business School, all of whom had taken advanced courses in probability and statistics.

There is a certain logical problem with saying that Bill is more likely to be an account who plays jazz, than he is to play jazz. The conjunction rule of probability theory states that, for all X and Y, P(X&Y) <= P(Y). That is, the probability that X and Y are simultaneously true, is always less than or equal to the probability that Y is true. Violating this rule is called a conjunction fallacy.

Imagine a group of 100,000 people, all of whom fit Bill's description (except for the name, perhaps). If you take the subset of all these persons who play jazz, and the subset of all these persons who play jazz *and* are accountants, the second subset will *always* be smaller because it is strictly contained within the first subset.

Could the conjunction fallacy rest on students interpreting the experimental instructions in an unexpected way - misunderstanding, perhaps, what is meant by "probable"? Here's another experiment, Tversky and Kahneman (1983), played by 125 undergraduates at UBC and Stanford for real money:

Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequences of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you chose appears on successive rolls of the die. Please check the sequence of greens and reds on which you prefer to bet.

1. RGRRR2. GRGRRR3. GRRRRR

65% of the subjects chose sequence **2**, which is most representative of the die, since the die is mostly green and sequence **2** contains the greatest proportion of green rolls. However, sequence **1** *dominates* sequence **2**, because sequence **1** is strictly included in **2**. **2** is **1** preceded by a G; that is, **2** is the conjunction of an initial G with **1**. This clears up possible misunderstandings of "probability", since the goal was simply to get the $25.

Another experiment from Tversky and Kahneman (1983) was conducted at the Second International Congress on Forecasting in July of 1982. The experimental subjects were 115 professional analysts, employed by industry, universities, or research institutes. Two *different* experimental groups were respectively asked to rate the probability of two different statements, each group seeing only one statement:

- "A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983."
- "A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983."

Estimates of probability were low for both statements, but significantly lower for the first group than the second (p < .01 by Mann-Whitney). Since each experimental group only saw one statement, there is no possibility that the first group interpreted (1) to mean "suspension but no invasion"*.*

The moral? Adding more detail or extra assumptions can make an event *seem more plausible,* even though the event necessarily *becomes less probable.*

Do you have a favorite futurist? How many details do they tack onto their amazing, futuristic predictions?

Tversky, A. and Kahneman, D. 1982. Judgments of and by representativeness. Pp 84-98 in Kahneman, D., Slovic, P., and Tversky, A., eds. *Judgment under uncertainty: Heuristics and biases.* New York: Cambridge University Press.

Tversky, A. and Kahneman, D. 1983. Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. *Psychological Review*, **90:** 293-315.

There's also a linguistic issue here. The English "and" doesn't simply mean mathematical set theoretical conjunction in everyday speech. Indeed, without using words like "given" or "suppose" or a long phrase such as "if we already know that", we can't easily linguistically differentiate between P(Y | X) and P(Y, X).

"How likely is it that X happens and then Y happens?", "How likely is it that Y happens after X happened?", "How likely is it that event Y would follow event X?". All these are ambiguous in everyday speech. We aren't sure whether X has hypothetically already been observed or it's a free variable, too.

In my experience, the english "and" can also be interpreted as separating two statements that should be evaluated (and given credit for being right/wrong) separately. Under that interpretation, someone who says "A and B" where A is true and B is false is considered half-right, which is better than just saying "B" and being entirely wrong.

Though, looking back at the original question, it doesn't appear to use the word "and", so problems with that word specifically aren't very relevant to this article.