Sunday, May 8, 2011

Q: Do the odds in a horse race add up to more than 100%?

It doesn't quite make sense to sum odds. But if we talk about converting the odds into the probabilities of each horse's winning, then yes, they do add to more than 100% -- because the house takes more than 16% of every dollar bet!

For example, let's say we had two horses in a race, equally favored to win. A "fair" race chart, with no house take, would give each horse 1:1 odds against winning -- meaning a successful $1 bet will pay back a total of $2. The probability that corresponds to 1:1 odds is 1/(1+1), or 50% -- and two of these sum to 100%. Say you want to be guaranteed to walk away with $1. Then you need to bet 1/2 dollar on each horse. Exactly one horse will win (paying off 1:1, so you'll get a dollar back), so you will break even.

In reality, the odds won't be 1:1 for each horse. It will be something like 2:3 odds for each horse, meaning a successful $3 bet will pay back a total of $5. The probability that corresponds to 2:3 odds is 3/5, or 60%. Two of these sum to 120%! Say you want to be guaranteed to walk away with $5. Then you must bet $3 on the first horse, and $3 on the second horse. You've bet $6 to be assured of winning $5 -- the house has taken 16.7%.

We can see this if you look at the chart from yesterday's running of the Kentucky Derby (http://www1.drf.com/tc/kentuckyderby/2011/pdf/2011-kentucky-derby-chart.pdf ):

The race went off like this:
  • Animal Kingdom, 20.90 (odds $1)
  • Nehro, 8.50
  • Mucho Mucho Man, 9.30
  • Shackleford, 23.10
  • Master of Hounds, 16.80
  • Santiva, 34.70
  • Brilliant Speed, 27.90
  • Dialed In, 5.20
  • Pants On Fire, 8.10
  • Twice the Appeal, 11.90
  • Soldat, 11.90
  • Stay Thirsty, 17.20
  • Derby Kitten, 36.30
  • Decisive Moment, 39.30
  • Archarcharch, 12.50
  • Midnight Interlude, 9.60
  • Twinspired, 32.90
  • Watch Me Go, 33.60
  • Comma to the Top, 35.80

Let's say on each horse we made a bet "to win" in the amount of what it took to be assured of walking away with $1. Exactly one bet will succeed, so the amount we need to bet is the reciprocal of one plus the odds. (E.g. if the odds are 9.60 against 1, we can bet 1/10.60 dollars to receive $1 if it's a success.)

The total we need to bet is

1/21.90 + 1/9.50 + 1/10.30 + 1/24.10 + 1/17.80 + 1/35.70 + 1/28.90 + 1/6.20 + 1/9.10 + 1/12.90 + 1/12.90 + 1/18.20 + 1/37.30 + 1/40.30 + 1/13.50 + 1/10.60 + 1/33.90 + 1/34.60 + 1/36.80 = 1.195...

So it takes about $1.20 to be assured of walking away with exactly $1. The house is taking about 16%.

(To be fair, the house is not just one house -- OTB facilities take a cut, etc. etc. But from the perspective of the gambler, the house advantage in horseracing is much higher than even the worst casino games.)

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