# Probability Question: Would A always have a greater chance of $A\cap B$?

My professor assigned a completely random question on our problem set. Basically, it goes:

Sophie is 30 years old and majored in philosophy. As a student, she participated in anti-nuclear demonstrations and was deeply concerned with social justice. Which is more probable?

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

I'm guessing the point the prof was trying to get across is that assuming you have two different events $A$ and $B$, $A\cap B$ would always have a smaller probability than either. Is this correct?

• This is the classic example used to demonstrate the conjunction fallacy. – Ilmari Karonen Sep 8 '12 at 23:50
• Your "greater" and "smaller" need to be non-strict; it is possible that the probabilities are equal. For example, if you pick a uniformly distributed real number in $(0,10)$, the probability of hitting something less than $5$ is exactly the same as hitting something less than $5$ and different from $\pi$. – hmakholm left over Monica Sep 9 '12 at 12:36

Yes. That is the idea. Since $A$ contains $A\cap B$ then $P(A)\geq P(A\cap B)$.

Mathematically, yes.

However... It seems to me that the reason so many people answer (2) is not just that they don't understand probability (that is true, but not the entire reason), but that the alternative means something different to them than to a maths professor. If you ask people what they assume the prof meant, they will predominantly say the alternatives were "Linda is a bank teller and not a feminist" vs. "Linda is a bank teller and a feminist", because that is the way normal people phrase their alternatives. In real life, nobody ever gives you two alternatives, one of which is a proper subset of the other. And for this interpretation, the answer may very well be correct.

Mathematicians may claim that the question means one particular thing, but as a linguist I sympathize with the view that expressions mean whatever the huge majority of speaker/hearers take them to mean, and that the question is intentionally misleading to make a point. (Which isn't even necessary, as I said - people are quite bad enough at probability that you could demonstrate it even with a fairly worded alternative.)

Yes. $A\cap B\subseteq A$, always.

The "point" probably is that a ridiculously large number of people choose $(2)$. You can fool enough of the people all of the time.

There is yet another interpretation of the question, which might be what actually motivates a large number of people to choose the second answer. The phrase "which is more likely" invokes the related, but formally distinct question of "in which scenario is the previous event more likely". The answer to this intuitively seems to be the second one.