The usual probability story in this context is something like the following. A New Jersey woman, Evelyn Adams, won the lottery twice within a span of four months raking in a total of 5.4 million dollars. She won the jackpot for the first time on October 23, 1985 in the Lotto 6/39 in which you pick 6 numbers out of 39. Then she won the jackpot in the new Lotto 6/42 on February 13, 1986. Lottery officials calculated the probability of this as roughly one in 17.1 trillion, which is probability that one preselected person won the lottery on two preselected dates.
When one realizes that (i) somebody won the October 23, 1985 lottery. (ii) We would have been equally impressed if this happened twice within a one year period. (100 twice weekly drawings) (iii) Many people who play the lottery buy more than one ticket. Taking these three things into account he probability ends up to be about is now about 1/200. If we take into account the number of states with lotteries. For more examples of things that aren’t as surprising as they seem look at http://www.math.duke.edu/~rtd/Talks/Emory.pdf.
A recent paper on the arXiv:1503.02902v1 by Rich Arratia, Skip Garibaldi, Lawrence Mower, and Philip B. Stark tells a different type of story. In Florida’s Play 4 game you pick a four digit number like 3782 and if all four digits match you win $5000. The fact that this event has probability 1/10000 and hence nets you 0.50 average, says either that (i) people can’t think or (ii) they have utility functions that value a large sum disproportionately more than the $1 you use to play the game.
Some people however are very good at winning this gamble. An individual that we will call LJ has won 57 times. Now that by itself is not proof of guilt. If he bought 570,000 tickets he would end up with about this many wins. However that seems a little unlikely. If he only bought 250,000 tickets the probability of 57 wins is 1.22 x 10-8. (Exercise for the reader.)
Arratia et al give a very nice calculation that shows something funny must be going on. Skipping the math, the bottom line is that if the 19 million people that live in Florida all sold their houses, and took the $175,000 in proceeds (this is the average house value) and bought lottery tickets (reinvesting the winnings) until they ran out of money, the probability that someone would win 57 times or more is 1 in a million.
How did LJ get so lucky? Well there are three common schemes. (i) A clerk can scratch the ticket with a pin revealing enough of the bar code to be able to scan it to see if it is a winner. (ii) Sometimes a customer will ask the clerk if the ticket was a winner. If so the clerk may lie about the ticket being a winner and keep the money himself. (iii) Sometimes the winner may be an illegal immigrant or owe child support or back taxes, and will sell the ticket to an aggregator who pays half price for it and later claims the prize. This is a good scheme for people who want to launder money.
It would be nice if I could tell you that probability helped catch a criminal but at least it wasn’t involved in a miscarriage of justice like Sally Clark experienced. She was convicted of murder based on the calculation that the odds were 73 million to 1 against two of her children dying of what is called cot death in the UK.