## How to turn betting odds into probabilities

A colleague emailed me today, asking how to turn betting odds into probabilities, and reminded me that this is something I’ve been meaning to blog about for some time. To some readers, it will be obvious, but to non-gamblers, it won’t. Since my parents met in a Methodist church, this isn’t something that came naturally to me the first time around.

Let’s take the example of PortlandBet, whichÂ currently has Labor at \$1.67, and the Coalition at \$2.15.

If there was no profit margin to the bookmaker, the inverse of these odds would be the probability of winning. In other words, Labor’s chance of winning would be 1/1.67=60%, and the Coalition’s would be 1/2.15=46%.

All well and good, except that Portlandbet does have a profit margin, so these two probabilities add to 106%. That’s obviously silly, so we need to adjust both probabilties downward. We just do this by dividing by 106%, so the final probabilities are 60%/106%=56% and 46%/106%=44%. Now, the two probabilities add to 100%.

So where A and B are the dollar odds (eg. \$1.67 and \$2.15), the formula is:

Prob(A win)=(1/A)/(1/A+1/B)

If you have more than 2 candidates, you need to take account of them all, eg. with 3 candidates:

Prob(A win)=(1/A)/(1/A+1/B+1/C)

If there are aÂ large number of candidates, and you’re in a hurry, you can take a guess at the denominator. For most bookies, it’s 1.06 to 1.09.Â

And of course, a simple rule of thumb is that if your final probabilities don’t add to 100%, you’ve done something wrong.

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### 7 Responses to How to turn betting odds into probabilities

1. AM says:

I’ve been watching blogging academics and methodists’ sons searching for a term to describe the “bookmaker’s profit margin” for years now.
The word you want is “vigorish”, or “the vig” if you want to sound even less academic.
The take from the overround (the 106% total probability) could only be termed a “profit margin” in a tote system, where the payouts are determined after all the bets are in. In a fixed odds system like Portlandbet’s the eventual profit will approximately match the vig only because the prices are adjusted as bets are placed, and it will never exactly match because each indivual bet upsets the percentages.

2. Andrew Leigh says:

AM, I’m doing the naive thang, but you willÂ find both words in my academic papers. And yes, perhaps we should call the vig the ‘expected profit margin’ (since if prices are actuarially fair, equal amounts should be wagered on each candidate).

3. Many thanks, the margin had me puzzled I rang Portlandbet but they were very uninformative.

4. derrida derider says:

Actually in thin markets the size of the vig for SP bookmakers can vary substantially. I’ve often though that one strategy would be to use this variation as evidence of mispricing to identify arbitrage opportunities (for example, take advantage of differences in odds between caribbean and pommy bookmakers for cricket matches that don’t involve the WI and England). All it takes is a well set up spreadsheet.

But then you’re probably better off specualting in financial markets, where economic growth means the vig is usually modestly negative.

5. AM says:

Sorry Andrew – I just read my comment again and it was unnecessarily aggressive.
But it was late and this been irritating me for a long time.
Honestly, I don’t even know what a methodist is.

6. Kevin Cox says:

We have been discussing this problem in the area of football tipping competitions where the polls are the entries and the betting is the betting. The betting almost always appears to be closer than the tipping results. This data set might be a good one for you to test more formally than we have done because there are large samples.

A possible confounding issue is that polling and betting are samples from different populations. That is, betters are a different group than people who are polled. Another is that betters are probably a larger sample than the people polled because if you are a better but do not bet then you should be included in the sample size.

The other issue is time. If the polls say that the result is 55 to 45 on the day before the election then the betting odds are likely to be much greater than those converted from 55 to 45.

7. Andrew Leigh says:

AM, thanks for your note. No offence taken!

DD, we tested for arbitrage opportunities across 5 bookies in 2004, and couldn’t find any that yielded non-trivial profits.

KC, you may wish to be in touch with Sinclair Davidson about this. I think he was going to do some work, if he hasn’t already.