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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 has occurred.

Jacqueling Gagne is a 47 year old who retired from her job as a computer 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

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 this here. It is said here 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  5,000 to 1 for a "low-handicapper," 12,000 to 1 for an "average player" were lower than any others estimate. Using the odds 1/5000 and  Mathematca it is easy to compute the probability of 7 success using the binomial distribution with p = 1/5000 and n  =  65.  Doing this gives 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.

There has been a lot of discussion about this event. See in particular the discussion by Carl Bialick and the comments of his readers on the Numbers Guy Blog. Carl Bialik writes a column called The Numbers Guy for the Wall Street Journal and maintains this Blog.

We like the following remarks made by Professor Sechrest when he suggested this topic.

The accounts of

the statistical estimation procedures were quite interesting and represented something of a tour-de-force in calculating probabilities. What seemed to me to be odd, however, is that no one recognized, at least not explicitly, that the probability of such a record, once beyond a certain point, ceases to be interesting. One simply reaches a point at which it is evident that the woman has some special skill. Packing on further improbabilities does not provide any new information. It is my understanding that she came to the attention of people for her golfing prowess some time ago, i.e., before she reached ten, further supporting the conclusion that she has more than luck going for her.

I have a pair of loaded dice that I use for demonstrations from time to time. After I have rolled about six sevens out of, say, nine tries, everyone agrees that the dice are loaded. Going ahead to roll another six sevens does not

impress anyone as remarkable.

=Discussion

(1) Of course we only hear about unusual events so we should really have a bigger sample space. How could this be done for this event?

(2) Keith Devlin is reported to have said "The only justifible probability is 100% since the event happened. What do you think about this?

(3) What odds would you give for Jacqeuline getting a hole in one in her next round of play?