## Wednesday, May 8, 2013

### Calculating odds at the racetrack

My wife and I watched the Kentucky Derby this last weekend. She asked me a simple question that got me thinking. "How do they calculate the odds on the horses?" I told her, "Gosh, that sounds like the sort of question that John the Math Guy should be able to answer!"

So I pondered a while, scratched my head, and then scratched my beard. I almost got around to scratching my...but I had it basically figgered out, so I went online to see if my suppositions made any sense. Sure enough, I was right.

Before explaining how it is done, I'll give you a multiple choice test.

Q. How does a racetrack compute the odds for a horse race?

a) They determine the probability density functions of race time for each horse, based on previous performances, and corrected for the jockey and the condition of the track. These functions are then corrected for the horse's predilection to either go faster or slower when racing against other horses. Monte Carlo analysis is then used to simulate the probabilities of each horse winning.

b) They adjust the odds so that no matter which horse wins, the racetrack always get a certain percentage.

Disclaimer

Before I go on, I feel that I should share a bit about me, just so I don't give anyone the wrong impression. I am not a gambler, and I have never played one on television. I was in Vegas, once, for a business convention. I saw some good shows there, but the biggest bet I made was when I put three quarters in a machine, pushed the button, and got a can of Coke [1]. I figger I came out ahead. I was thirsty.

I did enjoy watching the race this last weekend, as I always do. The horses are beautiful, and the race is exciting.

Now for the "Truth or Dare" part -- my embarrassing public confession. The thing I like about the Kentucky Derby? You really wanna know? It's the hats. I love seeing women with huge hats. The bigger and more ostentatious the better. I just go weak in the knees.

Any caption under this picture would be funny

Back to the topic

Now that I have that off my chest, I will resolve the suspenseful multiple choice test. The answer is b). The game is rigged so that the racetrack can't lose. Ever. Even if there is a huge payoff on a long shot. Should you feel sorry for the track cuz they lost a lot of money? Nope. They make pretty much the same percentage regardless of the outcome.

So, just how do they figger?

Let's take an example. Let's say that the track brings in one thousand bets on a given race, each a bet of one dollar. [2] That means they have \$1,000 coming in. Each track has it's own "take", but let's just say that this particular track has a take of 15%. They will automatically pocket \$150 before doing any ciphering. This leaves \$850 for the payoffs.

Now let's just say that The Lovely Mare in the Big Hat received a total of \$200 in one dollar bets. The track has to consider, in the unlikely case that this nag actually makes it to the finish line without having to go back to get her hat, what can we afford to pay? The pot is \$850 dollars, and it has to get split 200 ways. That means that everyone who bet on The Lovely Mare in the Big Hat should get \$4.25 for each \$1 bet they made. That's how they set the payoffs. It's that simple.
The fillies in the lavender hats are always my favorites

Let's take another example... Let's say that I Can Barely Get Outa The Stable in the Morning has a grand total of four bets. Hmmmm.... an \$850 pot split evenly between four tickets? I would say that each ticket should  be paid \$212.50.

Aside from a bit of rounding (which is always done in the favor of the track) this is how the payoffs are calculated. No fancy math, no determining of probabilities, or looking at the weight to stride ratio of the forelocks and fetlocks. No Bayesian analysis or conditional probabilities or Markov chains.

But I left one little confusing thing out, the special lingo that is used to explain what the odds are..

What does "6 to 4 odds" mean?

Here is something I learned today. Odds are communicated in a funny way. Go figger. When someone says that the odds on a given horse are 6 to 4, it means that if the horse wins, I will have a profit of 6 dollars for every 4 dollars that I invested. I find that confusing. Of course nine out of every seven adults have trouble with fractions, but odds are no stated as a simple fraction.

I'll try to 'splain it again. Odds of 6 to 4 means that if I put 4 dollars in, I will get 10 dollars if I win. 4 dollars of that 10 will be returning what I put in for bet, and 6 dollars will be the payoff.

In the case of the \$850 pot, split 200 ways... I said that a \$1 bet should get a payout of \$4.25. In other words, a \$1 bet gets a profit of \$3.25. Thus, the odds would be 3.25 to 1. Maybe this would get reported as 13 to 4? Or maybe it would be rounded (in favor of the racetrack) to 3 to 1. I dunno.

Another thing that makes me go "hmmmmm"

I wonder...

I have made the point that the racetrack doesn't use any historical information on horses and jockeys to figure odds. They don't go through any fancy math to determine the probabilities that each horse will win.

Now that I said that, I'm going to disagree with myself.

Consider this. Let's say that the odds for Old Stewball [3] are hanging at 50 to 1. That means, of course, that a \$1 bet will return \$51 if you win. Now, the folks at the track will have been doing their analysis. Maybe some will look up Stewball's track record. Some will have a look at the jockey to see if he has fire in his eyes. Some will just say "What a cute name! My parents had that song on an album!" Based on their individual analyses, the folks at the track will put money on Old Stewball, or not. Essentially, they will use their dollar bills to vote on whether the odds are too low or too high.
Old Stewball might be just a bit long in the tooth [4]

The final odds are a crowd-sourced estimate of the probability that Old Stewball will win the race. The racetrack uses a sophisticated crowd-sourcing algorithm to figure the odds.

I find that pretty cool. Of course, I think the whole idea of doing crowd-sourcing is cool. One may think that crowd-sourcing is a new thing that came about only after the internet, but as I recall, there were racetracks around long before TCP met IP.

It makes me wonder... how accurate is this crowd-sourcing?  I will take a wild guess that any race that brings in a lot of professional gamblers will probably wind up with fairly decent estimates. Sounds like an interesting question to throw some data at.

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[1] A can of Coke from a vending machine for 75 cents?  Yes, this was a long time ago.

[2] Yeah, I know there's no such thing as a \$1 bet.  I was just trying to make the math more simple for all of you who aren't John the Math Guy.

[3] Bonus points to anyone who recognized this horse name from the song by Peter, Paul, and Mary or Woody Guthrie. If you caught this, send me an email (john@johnthemathguy.com) and I will email you back with one bonus point.

[4] The phrase "long in the tooth" actually came from horses. Unlike people, equine teeth grow their entire lives. If an old horse is not getting enough heavy chewing in, the vet will have to go in with a rasp and file down the back teeth. Honest. I've been the guy holding the tongue.