Conditioned thinking

Steve is a shy and tidy man. Is Steve more likely to be a librarian or a farmer?

 

With a few facts about librarians and farmers, it is possible to calculate the exact likelihood. In the US, there are twenty times more male farmers than male librarians. If four out of five librarians are shy and tidy and only one in five farmers have both those personality traits, then how likely is it that Steve is a librarian rather than a farmer?

 

Bayes' Theorem enables us to calculate the precise figure. The answer, perhaps surprisingly, is that Steve has only a seventeen percent chance of being a librarian rather than a farmer.

 

How can this be? If you guessed that Steve was more likely to be a librarian than a farmer then why does this intuitive answer contradict the calculated answer? The reason is that while we readily recognize the stereotypical personalities of librarians and farmers, we tend to ignore the general population sizes of the groups we are analyzing. The population sizes complicate the calculation, so we use a representative heuristic instead that neglects the relevant base rates.

 

The moral of the story is that we should try to identify base rate information and allow it to moderate our intuitive conclusions. In this case, the knowledge that there are many more male farmers than male librarians means that the answer is likely to be much less than Steve's resemblance to librarians suggests.

 

--

 

I came across the librarian story in Thinking, Fast and Slow by Daniel Kahneman. Working through the problem, we have:

 

  L = Librarian, S = Shy and tidy, P(L|S) is the probability of L given S

  P(L) = 1/20 [ the probability of being a librarian (actually 1:20 which is 1/21, but rounded for simplicity) ]
  P(S|L) = 4/5 [ the probability that a male librarian is shy and tidy ]
  P(S|~L) = 1/5 [ the probability that a male farmer is shy and tidy ]
  P(S) = P(S|L) * P(L) + P(S|~L) * P(~L) = 4/5 * 1/20 + 1/5 * 19/20 = 23/100 [ the probability of being shy and tidy ]

  Now plug the above values into Bayes' Theorem:
  P(L|S) = P(S|L) * P(L) / P(S) = 4/5 * 1/20 / 23/100 = 4/23 (17%) [ the probability that a shy and tidy male is a librarian ]