# Probability, Prosecutor’s Fallacy and Mr& Mrs Collins

There are three kinds of lies: lies, damned lies, and statistics.

-Benjamin Disraeli

In 1964 at Los Angeles, an elderly lady while walking down an alley saw a young lady with blond hair in pony tail running with her purse. From other end of alley a gentleman saw young white lady with blond hair in ponytail get into yellow car whose driver was black man with beard and moustache.

Few days’ later police arrested a couple- Janet Collins and her husband Malcolm Collins and charged them with theft. Though they met description, the elderly lady and gentleman could not positively identify them.

Then prosecutor decided to make use of statistics. He called a professor of statistics from local college. Professor explained rules of probability to jury. He first identified six features and treated them as independent events – black man with a beard, man with a moustache, white woman with blonde hair, white woman with a ponytail, interracial couple in a car, yellow car.

The prosecutor made some assumptions based on which he asked professor to calculate probability of couple being innocent. Probabilities based on assumption given below.

• Black man with beard: 1 in 10
• Man with moustache: 1 in 4
• White woman with pony tail: 1 in 10
• White woman with blonde hair: 1 in 3
• Yellow motor car: 1 in 10
• Interracial couple in car: 1 in 1000

Professor then used multiplication rule for independent events. So for ex. probability of finding black with beard AND man with moustache is calculated as…

P(black man with a beard AND has a moustache) = P(black man with a beard) x P( has a moustache) = 1/10 x 1/4 = 1/(10 x 4) = 1/40

So probability of a random couple satisfying above conditions was one in 1,20,00,000 i.e 1/10×1/4×1/10×1/3×1/10×1/1000=1/1,20,00,000.

Based on this it was concluded that chances of couple being innocent were 1 in 1, 20, 00,000 i.e. 1 in 1.2 crores. Based on these calculations jury declared couple guilty.

Couple went to Supreme Court. The Supreme Court reversed the judgement and freed the couple. The Supreme Court observed that … Mathematics, while assisting the trier of fact in the search for truth, must not cast a spell over him.

Supreme Court observed that professor made errors in calculations and misused concept of probability.

First, it was incorrect to assume six features as independent events ex. Black man with beard and man with moustache are not independent events. Men who grow beard also have moustache.

Secondly, what professor had calculated was probability of finding a couple matching description, but jury equated probability of match with probability of innocence- which is classical prosecutor’s fallacy. P (match) is not same as P (innocence)!

Thirdly going purely by probability in Los Angeles area it was possible to find one more couple meeting such conditions, so maybe it was other couple who committed theft and not this couple.