The Linda Problem

There is a famous problem first posed by the cognitive psychologists Amos Tversky and Daniel Kahneman called the "Linda Problem".  Here is the problem:

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.

Which is more probable?

1. Linda is a bank teller.

2. Linda is a bank teller and is active in the feminist movement.

Which answer do you think is correct?  If you chose option number two, you are in great company.  The vast majority (80% in fact) of study participants in Tversky and Kahneman's original study selected the second option, regardless of whether they were experts or not.  The "Linda Problem" is a classic case of what Tversky and Kahneman called a conjunction fallacy.  In lay terms, we fail to recognize in our minds the concept that the probability of two events occurring in conjunction is less than or equal to the probability of either one occurring alone.  Here is another example that might help make this point more clear:

Which of the following is more likely?

1. You will have a flat tire tomorrow morning.

2. You will have a flat tire tomorrow morning, and a man in a black care will stop to help you out.

Clearly, the second scenario is less likely to occur than the first one.  One of the fundamental laws of probability is that the likelihood of two events occurring at the same time (i.e., in conjunction) is always less than (usually) or equal to (possible, but still less likely) than the likelihood of either event occurring alone.  Written formally in mathematical symbols:  P(A^B) ≤ P(A) and  P(A^B) ≤ P(B).

Going back to the "Linda Problem", it just makes sense that someone who was "deeply concerned with issues of discrimination and social justice" and participated in anti-nuclear war demonstrations while in college would be active in these issues later in life.  Given no other career choice (both options said that Linda would become a bank teller), the second option seems more plausible, even though by the laws of probability and statistics it is not.

As I stated above, in Tversky and Kahneman's original study, even experts in statistics succumbed to the conjunction fallacy.  Here again we have support for Philip Tetlock's study published in his book, Expert Political Judgement and discussed in my last post, "Dart-throwing monkeys".  If experts in probability and statistics are fooled by the conjunction fallacy, the rest of us don't stand a chance!  That should at least provide us with some level of comfort (hey - even the experts fail to answer problems similar to the "Linda Problem").  It also should make us think twice before we make the same mistake.