An interesting application of simple probability theory came
when the Florida police department had to prosecute a defendant for buying an
illegal drug, cocaine. In the US, both buying and selling cocaine is a criminal
offense.
One night Florida police raided a suspicious facility, upon
finding nobody, they just seized the bags lying around. Out of the 496 bags
confiscated, they randomly chose four and found them to be cocaine which were
immediately destroyed (not the whole batch, just the four tested to be truly
cocaine). Come after few months, they randomly chose 2 packets from the lot and
sold it to a person posing as drug dealers. During the same night, they
arrested the person for crime of buying cocaine. They could not find the sold
packets in the person’s house and neither in his body (he tested negative for
drug intake).In the court the matter arose ‘What if the other packets were not
cocaine’, how could the person be prosecuted without any evidence. Florida
Police consulted the local statistician and asked the probability of the person
being innocent. Turns out the answer could be computed without much difficulty.
Out of the 496 packets, say N are packets of cocaine and
496-N are something else, say X (which is not illegal). The probability of the
person (henceforth defendant) being innocent is the probability of the event in
which the defendant bought X which would be the case corresponding to the
situation that Florida police chose 2 packets of X from 496-4=492 bags and sold
it to the defendant.
Since N could be any number from 1..496 we will try to estimate N. The court being an
authority of fairness would not convict the defendant without any strong
reasons. Shuster computed the maximum likelihood estimate of the given event
i.e. what must be the number N such that it maximizes the probability of
innocence, which is fairly easy to compute.
Maximizing P is same as maximizing log (P)
(since log is a 1-1 function).
This function can be numerically
optimized w.r.t N and since those days (which was in 1991) had good computers,
the result came out to be N=331 and the probability of such an event occurring
(by plugging the values back) is 0.022 which is a modestly lesser than the
probability of an event in which there are 100 men (including one criminal) and
you point out a random guy and proclaim him to be the culprit. Hence the defendant
was prosecuted by modest application of probability theory in real life.
References:
1.
Shuster, J.J (1991).The Statistician in a
Reverse Cocaine Sting. Amer. Statist, 42,203-204.
2.
Statistical inference, Casella and
Berger, 2nd edition, Duxbury Advanced Series.
