### Bayes’ Theorem and the Fake Facebook Lottery Winner

After the recent $587.5 million Powerball jackpot, the following picture was posted to Facebook, where it was shared over 2 million times. It is not too hard to figure out that the picture is a fake. The most obvious clue is that the numbers are out of order. The numbers on an authentic Powerball ticket are always in ascending order, except for the last number. But I was doubtful for another reason — Bayes’ theorem. There are two scenarios that must be considered. The first scenario is that Nolan actually did win the jackpot, and he wishes to share his good fortune with a random stranger. The second scenario is that Nolan did not win the lottery, and the picture is a fake. (Other scenarios are theoretically possible, but we will disregard them.) Let us define some events. Let A be the event that a player selected at random wins the Powerball jackpot, and let B be the event that a player selected at random would make an offer similar to the one shown above. We wish to estimate the conditional probability of A given B. According to Bayes’ theorem, the probability can be computed as follows: $P(A|B) = \displaystyle \frac{P(B|A) P(A)}{P(B|A) P(A) + P(B|\neg A) P(\neg A)}$ The probabilities in this formula are difficult to estimate, but let’s make an attempt. We know that there were two winners in the drawing, and I will guesstimate that about 20 million people bought tickets, so P(A) = 1/10,000,000. The other probabilities are more difficult to predict. It certainly seems unlikely that a lottery winner would offer to give$1 million to a complete stranger. Perhaps it’s even more unlikely that a (randomly selected) non-winner would pretend to win the lottery and offer to share it with a stranger. But it does not seem reasonable to suppose that the first event is 10 million times more likely than the second. So we must conclude that the denominator is dominated by its second term, hence P(A|B) must be close to zero. In other words, the picture is (probably) fake.