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Against all odds?

113,527,276,681,000,000 to 1
Those were the odds local golfer Jacqueline Gagne could make seven holes in one in 14 weeks. The Desert Sun
Palm Springs
Larry Gohannan
April 28, 2007

Lee Sechrest suggested this would be a good example to discuss how we should react to a claim that an event with an unbelievable small probabillity occurred.

Jacqueling Gagne is a 47 year old who retired from being computer progremer and moved to Rancho Mirage, California. Here she continued her love of sports by playing golf four or five times a week. Between January and May she made seven holes in one. In the Desert Sun article we read

Before she recorded her eighth this week, Professor Mike McJilton of the College of the Desert Math and Science Department calculated the odds of making seven aces in approximately 65 rounds (five rounds a week for 13 weeks) as Gagne did is just under 114 million billion to 1. As a decimal number, the chance of making the seven aces is 0.0000000000000000088

Another article said that McJilton used the poisson distribution presumeably as an opproximation to the binomial distribution. For this he needed an estimate for the probability that a reasonable golfer makes a hole in one in a round. There have been many attempts to estimate this. You can read about these here. This article says that the Golf Digest hired Francis Scheid, the retired chairman of the math department at Boston University, to calculate the odds using the latest and best information available. The odds Scheid came up with were lower than any others: 5,000 to 1 for a "low-handicapper," 12,000 to 1 for an "average player." Using these odds 1/5000 and Mathematca it is easy to compute the probability using the binomial distribution with p = 1/5000 and n = 65. Doing this we found the probability in agreement with McJilton’s result.

The next mention of calculating the odds was in the ESPN sports news May 24, 2007. By this time Gagne has made 10 holes in one in 16 weeks. Here we read

We asked one of the world's foremost mathematicians, Joseph B. Keller, professor emeritus of mathematics and mechanical engineering at Stanford, to compute the odds of making 10 aces in 80 rounds (using a previous finding by Golf Digest that the odds of an average player making an ace in any given round are 1 in 5,000). Keller's answer: "Roughly two chances out of 10 followed by 24 zeros. This is the same chance as picking one particular molecule out of all the molecules in 50 gallons of air."

By June 3 Jacqueline had made 14 holes in one in the past 5 months. Again using the Binomial model, the probability of this event is .0000000000000000000000000000000000003 Of course people were suspicious of a fraud but they were able to find witnesses to all of her hole in ones. Her authenticity gained further support when an ABC TV station sent a crew to interview her and she hit a hole in one while they were there. You can see this hole in one here.