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Beating the bookies: The maths of a World Cup 2010 win

Updated Friday, 2nd July 2010

The World Cup is a statistician's dream - but can you use maths to break a bookie's bank and heart?

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As a mathematician, the world cup looks a little different to me. It looks like a statistician’s dream with tables and tables of data in the papers and on TV. In fact, the whole World Cup is an exercise in data analysis.

So who will win 2010? Okay, not England, that's clear now. But is there anything we can salvage from the World Cup? Anything we can use at the bookies? Whilst examining the data I came across the notion of a Dutch Book. ‘In gambling a Dutch book .. is a set of odds and bets which guarantees a profit, regardless of the outcome of the gamble.’ (Wikipedia) Can we achieve such a thing? Win at the bookies if not on the field? We’ll need to understand a bit more about the odds before we can find out.

A betting slip from the 2010 world cup Creative commons image Icon Tim Snell under CC-BY-ND licence under Creative-Commons license


As of 12th June, Ladbrokes had odds of 8 to 1 (8 : 1) that England would win the cup. And 5.5 : 1 that Brazil will win. That means if you put £1 on Brazil to win and they do then you get £6.50 back, ie a profit of £5.50.

To see how the ‘book’ is made, let’s take an example. When a football match is to be played, there are three outcomes: Team A win; Team B win or they draw. Let us suppose the true odds are - 1:1, 7:3, 4:1 respectively. (NB. one way to think about a draw having a 4:1 chance of happening is that if we played the match five times, once there would be a draw.) The bookmaker will then offer odds along the lines of those given in the table below:


Table A Win

True Odds = 1:1

Bookmaker Odds = 5:6

Table B Win

True Odds = 7:3

Bookmaker Odds = 2:1


True Odds = 4:1

Bookmaker Odds = 7:2

By changing the odds in this way the bookmaker decreases the pay out he would make if he had offered the true odds. This is called the ‘overround’. It allows the bookmaker to make a profit over time. Individual events don’t guarantee profits for bookies.

Bookmakers have to protect themselves against large pay outs. This is realised by having balanced bets on all three outcomes, a ‘spread’ of bets. To achieve this they shorten the odds on the most popular bet and lengthen them on the less popular outcomes.,eg the odds change as the event draws closer.

In general people bet for one side or other to win. People don’t like to bet on a draw. Its just human nature. Because of this the odds on the outcome of a draw are lengthened out of proportion of the likelihood it will happen.

BETTER’S TIP NUMBER ONE: Place your bets on the draw.

Unless, of course, the tournament doesn’t allow for draws. Over time you will fair better. In realistic terms you will lose your money more slowly.

The odds are mostly determined by the amount of money that has been bet on a particular outcome. Determined by perception and psychology rather than a ‘rain man’ style data analysis. The odds of 8:1 for England told us more about the feelings of the English public than they did about England’s actual chance of winning.

To get a better sense of a country’s true odds we should look at what odds they are being offered abroad. This gave me an idea for how to make the Dutch Book. Bet abroad. Use what we know of international psychology and relations. Think about the Eurovision song contest. Think which countries give each other ‘nil point’ no matter what.

Betting abroad

We can now bet internationally using online companies. As Spain were the favourites we could expect a lot of money to have handed to the bookies there. This would be a good place to bet against Spain. This is where I hit the first hitch. Online betting companies seem to be largely owned by Brits (though they may be based in Malta or Gibraltar). So the odds do not differ as widely as we might hope.

All is not lost. There are still the betting exchanges. These are a relatively new and interesting phenomenon. Here bets are being accepted and offered simultaneously through the betting exchange's super computer.

The effect to the gambler is that you can place a bet at any odds that some other gambler is happy to accept. The odds that the betting exchanges offer vary widely. If you want to make a Dutch Book, your only hope lies with different international betting exchanges.

I leave this as an exercise for the reader. Answers on a postcard …

Like this? Check out 'Beating the odds: The maths of a World Cup 2014 win'.





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