Nebraska has a lottery. One of the games is “Pick 3”, where each player selects 3 one-digit numbers, and once a day a computer selects a winning sequence. The player wins if they have the same three numbers in the same order.
Monday (Jan 19, 2009), the winning numbers were 1-9-6. One lucky Nebraskan had bet the same combination, winning the top prize of $600.
Tuesday (Jan 20, 2009), the winning numbers were ALSO 1-9-6. This time, three people had the winning combination. (No, the person who won on Monday didn’t play the same combination on Tuesday.)
The news accounts of this event say that the two drawings were done on different computers (preempting the unspoken suspicion that there was some systematic error in the process).
According to the Associated Press, the probability of this happening are one in a million. And indeed, there are 1,000,000 possible combinations of winning numbers over two consecutive nights, since if you just concatenate the digits you get all the strings from 000000 to 999999.
But to say the probability is one in one million means you’re measuring the probability of having 1-9-6 come up a winner in both drawings. Most readers of the news story probably aren’t struck by that specific combination, but merely from the fact that the two numbers agree. That sort of coincidence is far more likely to occur, with a probability of 1 in 1000 (since there are 1000 possible draws on Tuesday, and only 1 of them will match Monday’s combination).
The probability that two consecutive draws do not match is 999/1000. But the probability that after n+1 drawings, no consecutive pair will match, is . For n=366, we find that the probability of no matching pairs to be roughly 70%, which means that over the course of one year of Pick 3 drawings, there’s a 30% chance of having the same numbers win two nights in a row at least once that year.
In two years of Pick 3 drawings, there is a more than 50% likelihood that the winning numbers will match on some pair of consecutive evenings.
As one of my Philosophy professors used to say, “Some surprises are not unexpected.”