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There is a very popular fallacy among sports betters that, if you want to bet on TWO results, then the ‘odds’ you get should simply reflect the multiplication of the two separate probabilities.

Suppose, for example, that you want to bet on horse X winning race one AND horse Y winning race two.

Suppose that

- the probability of horse X winning race one is 0.2
- the probability of horse Y winning race two is 0.5.

Using the ‘odds’ way of representing probabilities (and ignoring the bookies’ mark-up of the ‘odds’) then the separate bets would be 4 to 1 and evens respectively and the ‘double’ bet would be 9 to 1. Whichever way you look at it, if you bet one pound then you will get ten back.

This seems like a sensible way to calculate the odds of the 'double' event.

But, the only time you can simply multiply the probabilities together to calculate the correct probability of the ‘joint’ event is if both events are independent. This means the outcome of the second event in no way depends on the outcome of the first. For the horse races (assuming all the horses in the two races are different) this might be a reasonable assumption. But for many sporting doubles it will not be.

Think of the following extreme example.

- The probability of Spurs drawing a particular football match is 0.2 (odds of 4 to 1)
- The probability of Arsenal drawing a particular football match is 0.2. (odds of 4 to 1)

In fact, in most cases it turns out bookies calculate double odds in a way that benefits them by capitilising on fans wrongly assuming events are independent. Consider the following example

- The probability of Spurs winning a particular football match is 0.4
- The probability of Arsenal winning a particular football match is 0.2.

It turns out that every day punters are seduced by apparently attractive odds on double bets which are actually nothing less than a rip-off. The classic example of such bets in football is the “Joe Bloggs scores AND Team X wins” variety.

For example, in March 2007 Spurs were playing West Ham. Based on odds offered by bookies the probability of Spurs winning was just over 0.5. The probability of the West Ham player Tevez scoring was 0.1 (he had never previously scored a goal in English football).

A good friend of mine confidently predicted that Tevez would score but that Spurs would win.

Multiplying the two probabilities together, the bookies would have given my friend odds representing a probability of 0.05 for the double bet. That looks good -- 20 pounds won for each pound bet. But he would have been massively cheated by this. The real probability of the double (a prediction which did in fact come true) was about 0.02, and so the odds he should have got were about 50 to 1.

It is all because the two events are definitely NOT independent. If we know that Tevez scores, then the probability that Spurs win is significantly reduced. In fact the following assumptions are reasonable:

- Probability Spurs win GIVEN that Tevez scores is 0.2
- Probability Spurs win GIVEN that Tevez does not score is 0.55

The real probability that we need to calculate is the probability that Tevez scores AND that Spurs win given that Tevez scores. This probability is equal to 0.1 times 0.2, i.e. 0.02.

The above argument has introduced you discretely to the idea of conditional probability.

Please note: Much of the material here has now been incorporated (and improved) into our new book "Risk Assessment and Decision Analysis with Bayesian Networks" by Norman Fenton and Martin Neil. The book includes many more worked examples like these. The website for the book is here while the book's associated blog is here. Also, we now have a regularly updated blog addressing similar themes: Probability and Risk.

Norman Fenton

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