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I was reading this article which has the following question,

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?

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

I chose the second option but according to a set theorem the choice was wrong. Why is not possible that linda can be both a bank teller and feminist? What is the simple explanation to this with example?

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Of course any discussion of the conjunction fallacy should include both an explanation of why it is a fallacy and an explanation of why people make it. My guess is that we have evolved to trust stories with more (sensible) details, since most of the stories we hear come from other people and details are harder to lie about. –  Qiaochu Yuan Apr 20 '12 at 6:10
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Which is more likely: (a) it's raining where I live, or (b) it's raining where I live and I'm eating toast? –  Billy Sep 14 '12 at 9:38
    
the former I guess...assuming no premise is given. but probability is non intuitive. is there anything that I missed out in the premises? –  kingboonz Sep 14 '12 at 9:39
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The premises are irrelevant. All that matters is that it’s harder for (II) to be true than for (I) to be true: if (II) is true, then (I) and something else unrelated are both true. Thus, (II) must be less likely. –  Brian M. Scott Sep 14 '12 at 9:45
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No, you didn't miss anything. Obviously, the more conditions we attach to an event, the less likely it is to happen. (Strictly speaking, we don't know whether (II) is less likely than (I) or (II) is just as likely as (I). But, practically speaking, the latter is unrealistic... unless we think that, because she is active in the feminist movement, she must work in a bank for some reason.) –  Billy Sep 14 '12 at 9:52
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4 Answers 4

up vote 6 down vote accepted

Let $A$ be the event "she is active in the feminist movement" and let $B$ be the event "she is a bank teller." Then "she is active in the feminist movement and is a bank teller" is $A\cap B$. Whatever opinions one might have about whether bank tellers might or might not have a tendency to be active in the feminist movement, it is for certain true that $A\cap B \subseteq B$. Thus it is automatic that $$P(A\cap B) \le P(B).$$ So we cannot have $P(A\cap B) \gt P(B)$. It is logically impossible. Information that we are provided about Linda's background and history cannot alter that fact.

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If the question was (mentioned at bottom of article), There are 100 persons who fit the description above (that is, Linda’s). How many of them are: 1.) Bank tellers? __ of 100 and 2.) Bank tellers and active in the feminist movement? __ of 100. Then I say, 10 of them are bank tellers and 90 of them are bank tellers and feminists. Would it mean all of them are bank tellers? –  user102351 Apr 20 '12 at 6:06
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@user102351: that is impossible. Everyone who is a bank teller and a feminist is also a bank teller. "Bank teller" does not mean "only a bank teller." –  Qiaochu Yuan Apr 20 '12 at 6:07
    
So, all the 100 persons are considered bank tellers, and 90 of them are both bank tellers and feminists. –  user102351 Apr 20 '12 at 6:47
    
@user102351: you don't know how many of them are bank tellers, but sure, that's a possibility. –  Qiaochu Yuan Apr 20 '12 at 6:53
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I'm not sure if this answer is appropriate here, since it doesn't concern mathematics. It certainly concerns the question though.

This question is essentially a modified version of one given by Kahnemann and Tversky in a series of experiments. Even tough the probability that both A and B are true can never be higher than the probability of A alone, many test subjects violated this basic rule of probability. They commitet the conjunction fallacy.

The explanation of Kahnemann and Tverskywas based on people using a certain heuristic. People picture the situation described, and the easier it is to picture the situation, the more probable they consider the situation to be. Since the more extensive description paints a clearer picture, it gets associated with a more probable event.

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All the information about Linda's background is a complete red herring, intended to distract you into a wrong answer.

The simple fact is that as you add more constraints into a proposition, the circumstances in which that proposition is true can only decrease. More conjunction terms strengthen a proposition. They make it more specific.

This works across logic and set theory.

The intersection of sets A and B cannot be any larger than set A.

The stronger proposition P and Q cannot be true in more circumstances than the weaker proposition Q.

The number of women who are bank tellers and active in the feminist movement can be no larger than the number of women who are bank tellers (or the number who are active in the feminist movement).

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How does this add to Andre's earlier answer? –  Gerry Myerson Apr 20 '12 at 6:15
    
Earlier? In what frame of reference? I started typing this some half and hour ago, then was distracted away. Looking at both answers, I see that mine uses very plain language. What does that other one add to mine? –  Kaz Apr 20 '12 at 6:19
    
Earlier, in that, as I'm looking at the screen, under your answer it says "answered 7 minutes ago," whilst under the other it says "answered 35 minutes ago." What does it say on your screen? –  Gerry Myerson Apr 20 '12 at 6:22
    
The same: answered N minutes ago, and N + ~28 minutes ago. –  Kaz Apr 20 '12 at 6:24
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I think that reiteration of basic principles, particularly using somewhat different language, adds to the information made available to the OP. –  André Nicolas Apr 20 '12 at 6:56
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Edit: because of a question merge, it should be clarified that the options A,B,C are referring to the propositions

  • (A) It is more likely that "she is a feminist" than "she is a feminist and works in a bank"
  • (B) It is less likely that ....
  • (C) It is equally likely that ....

C can only be true if the chance of her being in an activist movement is 0% or the chance of her working in a bank is 100%.

B can only theoretically be true, if the chance of her working in the bank is more than 100%.

Edit: the previous answer was actually invalid (sorry!): C can only be true if the chance of her working in a bank is 100%.

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THANK YOU~~~~~~~~~~~ –  kingboonz Sep 14 '12 at 10:16
    
No, C is true if the probability of her working in a bank given that she is active in the feminist movement is 100%. It doesn't have to be 100% overall. –  Arthur Sep 14 '12 at 10:24
    
That is essentially the same thing, because in both cases she is in the activist movement. –  Gideon Potgieter Sep 14 '12 at 10:58
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