Jumping to cognitative illusions
Below two excerpts from an interesting talk on Ockham's Razor.Imagine a lady who is small, shy, neat, and interested in detail. Is this person more likely to be a politician, a farmer, or a librarian? Easy! It is highly likely you thought this person is a librarian. But stop and consider how many politicians, farmers, and librarians there are in the world! It is actually much more likely that this person is a farmer. You have 'jumped' to a probability conclusion and over-estimated the likelihood of a stereotypical librarian. But you are not alone. Only one person from among the few hundred to whom I have posed this problem, including senior students at high school and university, as well as numerous trained medical researchers, has rapidly given me the correct answers, together with the correct reason.
Suppose there was a stabbing outside a nightclub. In court, the one eyewitness testifies that the assailant fled in a silver-coloured taxi. On the night of the offence, it is known that only 15% of taxis on the road were silver. Furthermore, when the crime scene is recreated, it is established that the witness is 80% accurate at picking silver from non-silver taxis. You are the judge. What is the probability that the taxi involved in the crime was silver? Initially, it might seem, given the eyewitness is 80% accurate, that the probability the taxi was silver, as claimed, is also 80%. But this ignores the error the witness makes when observing the much more common, non-silver taxis. In fact, to work out the correct probability, we need to invoke a theorem devised by the Reverend Thomas Bayes, published in 1763, two years after his death. This theorem allows probabilities to be calculated accurately on the basis of full knowledge of all initial possibilities. When this non-intuitive, but mathematically simple, theorem is applied, the true probability that the taxi at the crime was silver is found to be only 41%; less than a one in two chance. In the Australian tradition, you should therefore bet that the taxi at the crime was actually not silver. We jump to the wrong conclusion unless the Reverend Bayes' approach is applied.
posted by Mark Smalley @ 8:43 PM
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