What are the odds of Norm Simmons wearing two pairs of socks?

This from the Wall Street Journal about hitting multiple Hole-In-Ones




THE NUMBERS GUY By CARL BIALIK


Just How AmazingIs Jacqueline Gagne -- And Her 10 Golf Aces?May 18, 2007; Page B1
Jacqueline Gagne has had 10 once-in-a-lifetime experiences in less than four months.
Since Jan. 23, the 46-year-old from Rancho Mirage, Calif., has hit 10 holes in one, or just eight fewer than were hit on the entire Ladies Professional Golf Association tour last year.
Her local paper, the Desert Sun of Palm Springs, Calif., has corroborated Ms. Gagne's feat, running notes alongside articles from editors saying they're just as skeptical as readers, but everything has checked out.
The paper also asked a local statistician, Michael McJilton of the College of the Desert, to compute the odds against the feat. The result, which headlined the article: 113,527,276,681,000,000 to 1. And that was after just seven aces. I asked Mr. McJilton to repeat the computation after Ms. Gagne hit three more in the following couple of weeks, over a total of just 75 rounds. He returned the astronomical number of roughly 12 septillion (12 followed by 24 zeroes) to 1. Such an unlikely event should never happen. It's like winning the lottery four straight times. No wonder David Letterman came calling.

Is there an explanation for Ms. Gagne?
The numbers deserve scrutiny, and her feat isn't as improbable as it seems, though the odds against it are very steep. Probability may not explain it.
Mr. McJilton -- who says that as the only statistician at a local college, "I got the call" -- was using the Desert Sun's assumption that it takes the typical amateur golfer 5,000 rounds to hit a hole in one -- or 20,000 par-3 holes, since there are typically four shorter holes per round, and acing longer holes is so rare it can be ignored.
That estimate turns out to be pessimistic. There are data on millions of attempts at holes in one from events awarding prizes for aces. Two insurance companies that specialize in writing coverage for these basically agree on the odds for the typical amateur: Once every 12,750 shots on par 3s, says Mancil Davis, director of golf operations for National Hole in One Association; 1 in 12,500, according to US Hole In One.
The former, more-conservative estimates translates to roughly a hole in one every 3,200 rounds. That, in turn, leads to odds of 133 sextillion to 1 against, about a 99% reduction from Mr. McJilton's 12 septillion.
Mr. McJilton also assumed that no two holes in one could come in the same round, a safe assumption for most mortals but not for Ms. Gagne of late (though she hasn't yet doubled up). Allowing for magic striking twice in 18 holes, as David Boyum, co-author of the book "What the Numbers Say," suggests, reduces the odds against by a further 38%.
Still Ms. Gagne persists in defying the numbers. She has a seven handicap, better than the typical amateur. But the democratizing feature of a hole in one is that it isn't that much more common for Tiger Woods to ace than for an average duffer. The average winner of a hole-in-one prize is a 17 handicap, says Mr. Mancil. Pros hit holes in one about once every 3,750 par 3s, while the best amateurs ace one of every 7,500 -- half as often.
Let's say we apply the latter rate to Ms. Gagne (though Mr. Mancil says she's not quite yet in that group). We'd still get odds of 670 quintillion to 1 against.
How about the holes she has aced? Eight of the 10 are between 100 yards and 140 yards, but shorter holes aren't as easy as you might think. Both insurance companies say their rates rise by 10%, at most, for shorter holes.
There are three possible qualitative explanations for the feat. One, that the feat didn't happen, fits the principle of Occam's razor -- the simplest explanation tends to be best. But Ms. Gagne's story has checked out, with witnesses, in the Desert Sun's investigations.
Another is the notion of a hot hand, popular among basketball fans but not among researchers. They have found little evidence of streakiness as a factor in predicting whether the next shot will go in -- or how long any streak could last. Ms. Gagne's 10th ace caromed off a tree before sinking into the hole, which makes it neither illegitimate nor explainable by a hot streak.
Forty years ago, Mr. Davis had a hot streak: three aces in a week, five in a month and eight in a year. He has 51 career holes in one, considered by some the world record. He offers a third explanation: He aims for the hole. "Maybe because I made several early in my career, my brain focuses a lot better on the ultimate target," he says.
Mr. Davis says doctors measured his brain activity at the tee for par 3s and found it similar to that of golfers staring at long putts. He and some researchers also interviewed hundreds of Texas golfers, none of whom said they aim at the hole. Nor do most pros, who instead set up the best birdie putt.
Yet, Mr. Davis had an 11-year drought before hitting his 51st last month, at age 52. It made news before Ms. Gagne stole headlines. Ms. Gagne, meanwhile, says she just missed her 11th ace in four months by mere inches last week. "The only thing I thought about was the hole in one," she told me. "I didn't think of my surroundings. It was the perfect shot." Then, reconsidering, she adds, "If it was the perfect shot, it would have been a hole in one.