# Combinatorics and Probability Theory in Football Betting

## Introduction to Combinatorics and Probability Theory

This article is a step-by-step guide explaining how to compute the probability that, for example, exactly 4 out of 6 picks win, or how to calculate the likelihood that at least 4 of 6 bets win.

To help your understanding of this topic you will need to comprehend the basics of football result probability calculations, which I explained in detail in the article Calculation of Odds: Probability and Deviation.

### The Basics of Probability Computation in Football Betting

The following picks table contains 6 value bets including the calculated probabilities for each bet to win: English Premier League - Example Picks 22.3.2011

Of the 6 published picks, 4 won and made a profit of 19.9% on the 50.00 € betting bank. I will now attempt to explain the mathematics behind the above selections.

The calculation of the probability that all 6 Picks will win is relatively easy and requires no knowledge of difficult formulas. You simply multiply together the given probabilities, thus:

61.1% x 63.2% x 77.0% x 56.4% x 52.6% x 71.0% = 6.3%

The result of 6.3% is the probability that all 6 picks in the portfolio win.

Of course, the other end of the scale is that all 6 picks will lose. Again, this is a straight forward calculation: simply multiply the opposing probabilities to those used in the ‘win’ scenario, thus:

38.9% x 36.8% x 23.0% x 43.6% x 47.4% x 29.0% = 0.1973%

The result of 0.1973% is the probability that all 6 picks lose.

### Summary:

• Probability that all 6 Picks win: 6.3%
• Probability that all 6 Picks lose: 0.1973%

If you divide 6.3% by 0.1973% the result is 31.93. This means the probability in this particular portfolio that all 6 picks win is almost 32 times higher than the probability that all 6 picks lose.

Practically speaking, there is a 32 times higher chance of winning all 6 bets and cashing 40.90 € profit than losing all 6 bets together with the entire 50.00 € starting bank.

### Accumulated Betting Odds

• To win all 6 picks: 15.9 (1 divided by 6.3%)
• To lose all 6 picks: 506.7 (1 divided by 0.1973%)

These odds express that on average all 6 selected bets should win once in every 16 rounds and only once every 507th round should a total loss of the portfolio occur.

A single season’s football league betting will usually comprise approximately 80 rounds of matches (midweek and weekend betting). This means that statistically a total loss may happen once every 6.3 years betting on a similar portfolio to the example above each time. Of course, it could happen more often as wins and losses have a nasty habit of not lining up as cleanly as statistical theory says they should. For example, 2 total losses could occur in the first 2.6 years and then no more for another 10 years.

## What is the probability that exactly ‘X’ picks win or lose?

Further interesting questions include what are the probabilities that exactly 5 of the selected 6 picks win, or at least 4 of the picks win, and following this, it is natural to ask whether it is viable to make long-term profit on this type of portfolio and if so, how much?

An easy starting point for assessing whether a portfolio is ‘worthwhile’ is by calculating the ‘expectancy’, in other words, how many of the picks are likely to win. This is simply the average of the win probabilities of the selected picks:

(61.1% + 63.2% + 77.0% + 56.4% + 52.6% + 71.0%) / 6 = 63.55%

This value means that by betting on the above portfolio a success rate of 63.55% is ‘expected’, which would correspond to a hit rate of 4 from 6 picks (i.e. 6 [picks] times 63.55% = 3.81 [roughly 4 picks]). This means that on average this portfolio should usually bring around 4 successful picks. However, it is obviously necessary to check if the combination of 4 successful picks and 2 failed ones will produce a profit: Profit Calculation: Exactly 4 out of 6 Selections Win

The above illustration shows that every combination of 4 picks from our 6-match portfolio would have returned a profit of between 7.02 € and 16.71 € depending upon the combination.

### Important Note

Please note that the average value (expectancy value) does not mean a 63.55% probability that exactly 4 picks will win every betting round. The average value indicates that if you bet on this type of 6-match portfolio often enough, an ‘average’ of 4 hits can be expected.

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Last Update: 17 April 2015

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