Conjunction Fallacy

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.


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:

  1. “A com­plete sus­pen­sion of diplo­matic re­la­tions be­tween the USA and the Soviet Union, some­time in 1983.”

  2. “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.