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Solution: Bayes and Bobby Smith – in a nutshell.

April 3, 2019

Solution to the Exercise

We can solve the Lucy Jones problem using Bayes’ Theorem. The (posterior) probability that a hypothesis is true after obtaining new evidence, according to the a,b,c formula of Bayes’ Theorem, is equal to:

ab/ [ab+c(1-a)]

a is the prior probability, i.e. the probability that a hypothesis is true before the new evidence. b is the probability of the new evidence if the hypothesis is true. c is the probability of the new evidence if the hypothesis is false.

Before the new evidence (the test), this chance is 1 in 1000 (0.001)

So a = 0.001

The probability of the new evidence (the A+ score on the test) if the hypothesis is true (Lucy will become a professional player) is 100%, since all professional players score A+ on the test.

So b =1

The probability we would see the new evidence (the A+ score on the test) if the hypothesis is false (Lucy will not become a professional player) is 2%, since the test is 98% accurate in spotting future professional footballers.

So c = 0.02

Substituting into Bayes’ equation gives:

Posterior probability = ab/ [ab+c(1-a)] = 0.001x 1 / [0.001 x 1 + 0.02 (1 – 0.001)] = 0.001 / (0.001 + 0.01998) = 0.001 / 0.02098 = 0.0476 = 4.76 per cent.

So, using Bayes’ Theorem, the chance that Lucy Jones, who scored A+ on the test which is 98% accurate, will actually become a top player, is not 98% as intuition might suggest, but just 4.77 per cent.

An alternative method of solving this problem is to note that of the 1,000 children in total, in terms of probability 20 will be wrongly assessed as being future pro tennis players (the test is only 98% accurate), while only one of the thousand will in fact become a professional tennis player. Therefore the probability that Lucy will become a professional player if assessed as such is 1 in 21, i.e. 4.76 per cent.

 

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