Proving the Innocence of Phileas Fogg

By Data Tricks, 12 January 2018

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In Jules Verne’s novel Around the World in Eighty Days, Phileas Fogg is pursued by Fix, a detective from Scotland Yard in search of a bank robber who stole £55,000 from the Bank of England. From the description issued by Scotland Yard, Detective Fix mistakes Phileas Fogg for the bank robber, and a lengthy pursuit ensues while Fix awaits an arrest warrant from London.

The Prosecutor’s Fallacy

Given the lengths Detective Fix went to keep up with Phileas Fogg on his round-the-world trip, it would appear he was reasonably satisfied that Fogg was the culprit. After all, what are the chances of somebody from London fitting the description of a tall, well-built gentleman, about 40 years of age, bearded and with light brown hair? Let’s say that the probability of having each of those rather vague features in 19th century England were as follows:

Male = 0.5

Tall = 0.15

Well-built = 0.1

“About” 40 years of age = 0.08

Light brown hair = 0.17

Bearded = 0.3

The probability of somebody matching all of those characteristics is therefore 0.0000306. Or 0.003%. Perhaps Detective Fix had these slim odds in mind and concluded that there was a 99.997% chance he’d got his man.

This misinterpretation is commonly referred to as the Prosecutor’s Fallacy, where the probability of a match is confused with the probability of guilt.

Bayes’ Theorem

Bayes’ Theorem can be used to explain this:

$$P(A|B) = {P(B|A)P(A) \over P(B)}$$

Where:

P(A) = Probability of A

P(B) = Probability of B

P(A|B) = Probability of A given B

P(B|A) = Probability of B given A

In our example, we can say that event A is that Fogg is the bank robber. Event B is that Fogg’s appearance matches the description. Thus:

P(A) = 0.000000303 (given there were 3.3 million Londoners in 1873)

P(B) = 0.0000306

P(A|B) = Probability that Fogg is the bank robber, given he matches the description

P(B|A) = 1 (because if Fogg was the bank robber, there ought to be a 100% chance he’d fit the description)

Applying these numbers to Bayes’ theorem, we find that P(A|B) is actually 0.0099. In other words, the probability of Phileas Fogg being the bank robber is less than 1% – not 99.993% as first thought. Should Detective Fix have been schooled in statistics, perhaps he would have avoided the need to traverse the globe in pursuit of the innocent Phileas Fogg.

Footnotes

This is a simplistic illustration of the application of Baye’s Theorem using a fictional scenario and fictional values. There are, of course, other events that increased Fix’s suspicion, such as why Fogg left London in such haste.

Tags: Bayes, Bayesian, data science, Probability, Prosecutor's fallacy

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