Thursday, February 4, 2010

Randomness and probability: can people intuit probability?

One of my concerns with the evidence on the misinterpretation of randomness and probability in The Drunkards Walk arises during the discussion of the first law of probability: The probability that two events will both occur can never be greater than the probability that each will occur individually. While the law makes intuitive sense, the evidence that people fail to apply it to their reasoning is a little flimsy.

Mlodino asks us to consider an experiment from Khaneman, Slovic and Tversky’s famous book on judgement under uncertainty. Given the brief description about the Linda below, eighty eight subject were asked to rank the statements that follow on a scale of 1 to 8 according to their probability, with 1 representing the most probable, and 8 the least. The results are in the order ranked by the participants from most probable to least probable.

See the results under the fold and why their finding, that people have poor intuition of probability, may be incorrect.
Imagine a woman Linda, thirty-one years old, single, out-spoken, and very bright. In college she majored in philosophy. While a student she was deeply concerned with discrimination and social justice and participated in anti-nuclear demonstrations.

Statement Average probability rank
Linda is active in the feminist movement                            2.1
Linda is a psychiatric worker                                              3.1
Linda works in a bookstore and takes yoga classes              3.3
Linda is a bank teller and is active in the feminist movement 4.1
Linda is a teacher in an elementary school                           5.2
Linda is a member of the League of Women Voters              5.4
Linda is a bank teller                                                          6.2
Linda is an insurance salesperson                                       6.4

What stands out from this experiment is that the participants ranked the probability that Linda was a bank teller lower than the probability that she was a bank teller AND is active in the feminist movement. Since the bank teller statement captures the combined statement plus all options of being a bank teller and not active in the feminist movement, the probability should be higher.

My reason for questioning whether people actually fail to intuit the true probability is that my personal interpretation was that the Linda is a bank teller statement implied that she wasn’t active in the feminist movement. By default my brain registered the separate itemisation of the option to mean that it did NOT capture the bank teller and feminist statement.

In fact the respondents may have been completely correct if we apply the NOT criteria to the bank teller choice. Consider the diagram below. On the left is the way the researchers had the statement defined in their mind. On the right is the alternate interpretation where NOT feminist is implied as the first constraint in the sequence of reasoning. When we change our minterpretation of the statement, the probability of a feminist banker is 40%, while the probability of a non-feminist banker is a mere 10%.
So, depending on how we define the criteria, and how we perceive the problem, the probability changes.  My personal view is that people instinctively may be better at understanding probability than made out by Khaneman et al, and reiterated by Mlodino.

That being said, I currently have the difficult task of explaining why a mean rainfall and streamflow figure is a poor basis upon which to evaluate investment decisions for irrigated agriculture in catchments with highly variable rainfall.  One situation where people almost always fail to intuit probability is with weather variation. A weather event observed 10 times in a century becomes a 1 in 10 year event, which people then expect to happen with almost certainty every ten years.  In reality however, the chance that the 1 in 10 year event will occur in any 10 year period is closer to 65%, and for a twenty year period the chance of having a 1 in 10 year event is still only 87%.