Late on September 11, 2002, reporter Kate Sheehy at the New York Post phoned me for the answer to a probability question. (In retrospect, the reason why the Post phoned me, instead of someone in New York, is clear - they were betting that Hawaii is far enough away that the Post's track record w/r to accuracy might not have made it out here!)
The article that ran on September 12 appeared under the not-at-all-sensationalist headline:
LAST NIGHT'S CHILLING LOTTERY WINNER: 9-1-1
The article began:
Only in New York.
In a bizarre and eerie coincidence, the numbers drawn for the state's New York Numbers lottery game last night came up 9-1-1.
After some personal accounts which might or might not have been accurate, the article concluded:
A math expert said the odds of drawing numbers that match the date on which they're picked are not as long as might be expected.
Professor David Ross of the University of Hawaii said it’s likely to happen about three times a year.
"It's really not extraordinary," he said.
Of course, the quote is wrong. The actual estimate that I gave Ms. Sheehy was that the probability of a lottery number matching the date in the course of a year was around 3/10, which means roughly once in 3 years, not 3 times/year as reported.
That estimate was done off the cuff, and assumed that every date could be interpreted as a 3-digit number; in this case, the actual probability is 1-(.999^365)=0.306, so (unusually) I wasn't off by much. This leaves open the question of dates with 4 digits (eg, October 11 is 10/11); if we simply omit these 65 days/year, we get 1-(.999^300)=0.259, or about one match in 4 years.
I have just learned that New York runs this particular draw twice every day, not just once. It follows that the daily probability of at least one match on a given day is almost doubled, and the probability of a match some time in the year is either 1-((.999^2)^365)=0.518 if we include 4-digit dates, or 1-((.999^2)^300)=0.45 without them. Hardly chilling!
- David R.