Tuesday, October 16, 2012

Horse Racing and Coincidence

Just penned this to the BBC about why their piece on a jockey winning seven races in a single day quotes the wrong odds.
In your article on Richard Hughes winning seven races in a single day, you quote the odds of this event as being 10,168-1.  Whilst undoubtedly a fantastic achievement, these odds are incorrect, since they they ignore the fact that Mr Hughes raced in eight races that day.  The chances of him winning 7 out of 8 races (or more) is about 1,257-1, which is a bit more modest.  In particular, it seems fairly unlikely that someone would place a bet on the rider winning these particular seven races, and not the eighth.
If the piece's author or anyone else want to talk more about how these odds are calculated, or why this sort of thing is important, I'd be very happy to chat about it.
Best wishes,
Robin Evans
I think this is the right approach to take with these mistakes, and I hope it doesn't come across as condescending.  With a bit of luck they'll acknowledge the mistake and get in contact.

While we're on the subject, calculating multiple odds can seem tricky at first.  The quickest way to do it is this: if the odds an event are 'a to b' (usually written a-b or a/b), then the probability of the event is b/(a + b).  For example, 3/2 means 2/(3 + 2) = 0.4, so we'd expect 40% of events with these odds to actually occur (or slightly less if the bookies are taking a cut!).

To work out the chances of several events happening, and assuming these are independent (reasonable in this case, since the odds would be updated after taking into account what had happened in previous races), we multiply the probabilities.  In this case the seven races had odds of 13/8, 5/2, 7/1, 4/1, 5/2, 7/4 and 15/8, and multiplying these gives 0.000098, which as the BBC say is about 10,168/1.

However there was an eighth race, in which Hughes was the 2/1 favourite, and in which he 'only' came third.  So we need to consider the probabilities of him winning seven races, and failing to win the eighth race; of course there are eight ways in which he could do this, because we would have found it equally amazing if he'd won any of the seven races, and we throw in the chance of him winning all eight, since that would be seriously impressive.

The sum of all these possibilities comes to about 1257/1 (R code working below).  This is still impressive, but an order of magnitude different.  There's a lot of race meetings every year, so I'm surprised that this event hasn't happened since 1996 - perhaps jockeys don't often race eight horses in one day.  Or maybe their the odds on their horses are usually a bit longer.

There are plenty of other, examples where people straightforwardly fail to calculate odds correctly, making something seem strange, spooky or suspicious, when it's merely mundane or tragic.  It would be great if the BBC could avoid contributing to this malaise!

R Code

p = 1/(c(13/8, 5/2, 7/1, 4/1, 5/2, 7/4, 15/8, 2/1)+1)

out = prod(p)
for (i in 1:8) out = out + prod(p[-i])*(1-p[i])


1 comment: