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Bayes’ theorem and the base rate neglect

December 23rd, 2011 by dimpledbrain

This is a continuation of the previous posting. Please click here to go to the first posting.

The answer is 41%. Why?

To recap,

  • 85% of the car is Green and the remaining 15% Blue
  • A witness testified that the cab involved in the accident was Blue
  • The witness was tested for the ability to discriminate Green from Blue and was found to be correct 80% of the time
  • What is the probability that the cab in the accident was Blue as the witness testified?

(a)    80% (b)   50% (c)    41% (d)   29%

(1)    Probability that the witness correctly identifying a blue cab (15%*80% =12%)

(2)   Probability that the witness incorrectly identifying a green cab as blue (85%*20% =17%)

(3)    Probability that the witness correctly identifying a green cab (85%*80% =68%)

(4)   Probability that the witness incorrectly identifying a blue cab as green (15%*20% =3%)

Graphically depicted as follows:

Therefore, the probability that the cab will be identified as Blue is 29% (correctly identified as Blue 12%, incorrectly identified as Blue 17%).

Therefore the probability the cab identified as blue is actually blue is 12%/(12%+17%) = 41%

If you got your answer as either 80% or 50%, the above explanation should suffice.

If your answer is 29%, I offer you a supplementary explanation as follows:

Suppose that we have 100 such incidents. In 85 of these the taxi will have been green and 15 blue, just based on random selection of taxi colour. In the cases where the taxi was green the witness will mistakenly say that the car is blue 20% of the time, i.e. 17 times. In the 15 blue cases the witness will correctly say blue 80% of the time, i.e. 12 times. So although there were only 15 accidents involving a blue taxi there were 29 reports of a blue taxi being to blame, and most of those (17 out of 29) were in error. Now given that we were told it was a blue taxi, what is the probability that it was a blue taxi? That’s just 12/29 or 41.4%.

[the above explanation is courtesy of the Internet]

The above is called Bayes’ (pronounced Bays) theorem.

Salient points:

  1. If our answer is 80% i.e. based on what the witness says, we are ignoring the natural base rate effect. Think about the situations where people tell you they always win at 4 digit lottery or anything extraordinary. Base rate is 23/10,000 =0.23% (every 4 digits will give you 23 chances of winning from a pool of 10,000 combinations. 23 because there are 23 prizes, first prize, second prize, third prize, 10 starter prizes and 10 consolation prizes)
  2. If the above case is tweaked as 85% of the cars that are involved in the accidents are Green, your chance of calculating the probability will be more accurate. We have a tendency to link cause and effect. The point is under normal circumstances, we may tend to ignore the base rate effect.
  3. The next time someone quotes you some statistical numbers, always think about the base rate effect.
  4. We are no Bayes nor trained mathematicians so we don’t need to engage in the Bayes’ theorem calculation to arrive at the precise percentage. What we need to do at every instance is to make the base rate as our first referential point and to cross check with the incident we want examine.
  5. Be curious in almost anything so as to widen our knowledge on base rates.

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