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A one-in-17 million chance. Or was it?

Holes-in-one aren't as newsworthy as you'd think.

In October 2017, Jane Mattey and Clair Shine hit the headlines.  These two club golfers had done something extremely unusual: during a regular round at East Berkshire golf course, they had both made a hole-in-one at the same hole.

The headlines quoted it as a "1 in 17 million" coincidence, which meant it was up there with the chances of winning a big payout in the National Lottery.

There's a big difference between this and the Lottery, however, because while the odds in a Lottery are fixed and can be worked out exactly, the odds of a hole-in-one are more about sticking a finger in the air and making a lot of assumptions.

For a start, the odds of getting a hole in one depends on the player.  A top golfer like Rory McIlroy is far more likely to hit the pin than a regular club player.  

Then there's the length of the hole.  In order to get a hole-in-one, you need to be capable of hitting the ball from the tee to the green.  For almost all golfers this means holes-in-one can only happen on so-called Par 3 holes, which are typically between 100 and 200 yards long. The shorter the hole, the easier it is to get a hole in one because an error in the direction that you aim the ball is not punished so severely. 

There are usually four Par 3 holes on a golf course, so in a round of 18 holes, there are four opportunities to get a hole-in-one.  This means that the chance of getting a hole in one during a round of golf is about four times higher than the chance of achieving it on a particular, named hole.

The "One in 17 million" figure seems to have come from a US organisation called the National Hole In One Register. Based on stats that they have accumulated from around the world, they reckon that the chance of getting a hole in one is around 1 in 2,500 for a professional golfer, and about 1 in 12,000 for a club golfer.  So on a given hole, we might expect the chance of two average women golfers both getting a hole in one might be 1 in (12,000)^2.  That's more like 1 in 150 million.

But further investigation into the story reveals that the hole in question had been shortened to just 90 yards because the course was being repaired, which must have hugely reduced the odds of a hole in one.  And also, the women were part of a foursome, whom we will call A, B, C and D., This means there were six possible pairs of women who could have got a hole in one: AB AC AD BC BD and CD.  And it would have been a headline if any of those six pairs had been successful.  All of which brings our 1 in 150 million down to something much smaller.

In the end, nobody really cares if the odds are 1 in 17 million or 1 in 50 million: it's just a chance to report that something freakishly unlikely just happened.  But was it really freakishly unlikely?  Before I went in to record a radio item about this story, I thought I'd drop into my nearest golf club in Dulwich to see if I could pick up any anecdotes.  I spoke to the manager.  

"Holes in one? We get about ten of them a year.  In fact, we had an eleven year old get one last Sunday," and he showed me the card sitting on his desk.  "Oh, but if this is about those two ladies who got holes in one last week, we can beat that." He led me to a plaque on the wall outside his office. With a caption "Halved with holes in one!" it showed two smiling men who had just halved a hole in a matchplay tournament with holes in one.  This had happened in 1984.

So the first random golf club I turned up to was able to produce a more unlikely story than the 2017 ladies foursome story.

I made a quick back of envelope calculation: there are about 30,000 rounds of golf played at Dulwich each year, so about 100,000 par 3s are played.  In thirty years, that means there have been about three million opportunities for two people to get a hole in one at the same time, and it has happened at least once.  It suggests that odds of one in a few million are reasonable.  And since there are estimated to be over 500 million rounds of golf played around the world each year, we'd expect the story of the two ladies and the holes in one to be replicated several times each year.


p.s. To find the story of the double hole in one I googled "two women golf hole in one".  Sure enough, as well as the East Berkshire story, there was an almost identical story from Northern Ireland that had happened a couple of months earlier. This time it was Julie McKee and Mandy Higgins who had got holes in one as part of a foursome. The headline declared this was a one-in-a-million story. Those odds seem to be dropping rapidly.