Introduction Probability Myths Do's & don'ts Scams
 Tables
 Best games BJ strategy Odds table
 Betting
 Blackjack Casino Games Horse Racing Lotteries Online Casinos Poker Poker Machines Roulette Share Market
 Various

# Gambling related articles

### Part 2

By Scott McIntosh

### Introduction

In a previous article I described a model that uses the normal distribution to describe the outcome of AFL games.

Only 2 inputs are required: a mean (e.g. bookies line or handicapping prediction) and standard deviation (e.g. 38 obtained from historical data).

In this article I will describe how to use the model to determine different probabilities for a variety of outcomes. Please read Part 1 to understand the accuracy and limitations of this model.

### Method

For the following discussion, note that a winning margin can be positive or negative depending on which team wins. The mean / predicted line used for the following calculations is quoted as positive in terms of the favoured team's average winning margin. Therefore a positive winning margin represents the favoured team winning and a negative winning margin represents the underdog team winning.

The normal distibution is a continuous function (e.g. 1.2, -24.123, 39.5) whereas the final result of an AFL game is discrete, limited to integers (e.g. 1, -24, 39). To calculate the probability of a certain winning margin X occuring we use the normal distribution function to calculate the probability for the result to be between X - 0.5 and X + 0.5. For example for a winning margin of 39 we calculate the probability of the result occuring between 38.5 and 39.5. For the special case of the draw we calculate the probability between -0.5 and +0.5.

To calculate the chances of a team winning we could add the probabilities for a team winning by 1,2,3 ... up to infinity. In practice I use 999 or some such large number as the maximum possible winning margin. Instead of adding all the probabilities for each result together we can use what is called the cumulative normal distribution function (CNDF) to calculate the desired probabilities. Using the function notation CNDF(result, mean, standard deviation) we can determine the probability of a result falling between a minimum and maximum winning margin as follows.

 Probability (minimum, maximum) = CNDF (maximum, mean, standard deviation) - CNDF (minimum, mean, standard deviation) = CNDF (maximum, predicted line, 38) - CNDF (minimum, predicted line, 38)

Examples

1. Line 15.5 Favourite team to win
Probability (0.5, 999) = CNDF(999, 15.5, 38) - CNDF (0.5, 15.5, 38)

2. Line 15.5 Underdog team to win
Probability(-999, -0.5) = CNDF(-0.5, 15.5, 38) - CNDF(-999, 15.5, 38)

3. Line 15.5 Draw
Probability(-0.5, 0.5) = CNDF(0.5, 15.5, 38) - CNDF(-0.5, 15.5, 38)

4. Line 15.5 Favourite to win by under 39.5 points
Probability(0.5, 39.5) = CNDF(39.5, 15.5, 38) - CNDF(0.5, 15.5, 38)

5. Line 15.5 Underdog to win by over 39.5 points
Probability(-999, -39.5) = CNDF(-39.5, 15.5, 38) - CNDF(-999, 15.5, 38)

To calculate the precise values for these equations you can use the online calculator provided or an Excel spreadsheet as described below.

### Excel

Excel provides the very useful NORMDIST function which can be used to easily calculate probabilities as previously described.

The following cells are what a user would input.
Cell A1: Predicted Line
Cell A2: Standard Deviation
Cell A3: Minimum winning margin
Cell A4: Maximum winning margin

The following cells contain formulas to calculate the results. Cell A5 represents the probaility and Cell 6 represents the 'fair' odds given this probability.
Cell A5: = NORMDIST(A4, A1, A2, TRUE) - NORMDIST(A3, A1, A2, TRUE)
Cell A6: = 1 / A5

Examples
1) Line 15.5 Favourite team to win
A1 = 15.5, A2 = 38, A3 = 0.5, A4 = 999
A5 = 0.65348, A6 = 1.53026

2) Line 15.5 Underdog team to win
A1 = 15.5, A2 = 38, A3 = -999, A4 = -0.5
A5 = 0.33685, A6 = 2.96860

3) Line 15.5 Draw
A1 = 15.5, A2 = 38, A3 = -0.5, A4 = 0.5
A5 = 0.00966, A6 = 103.51707

4) Line 15.5 Favourite to win by under 39.5 points
A1 = 15.5, A2 = 38, A3 = 0.5, A4 = 39.5
A5 = 0.38965, A6 = 2.56640

5) Line 15.5 Underdog to win by over 39.5 points
A1 = 15.5, A2 = 38, A3 = -999, A4 = -39.5
A5 = 0.07389, A6 = 13.53236

### Accuracy of Predictions

Graph 2 indicates the frequency of actual versus predicted results using the bookmakers line as the mean for the normal distribution model. It examines the results for the over/under 39.5 point margin bet frequently offered by bookmakers. Although there does seem a minor bias toward home teams winning more frequently than predicted, the predictions correlate closely to reality.

### Graph 2 - Under/Over 39.5 Point Results (1056 Home and Away Games : 2000 - 2005)* *Note: Team 1 is the home team as listed in the fixture.

### Conclusion

Applying the normal distribution model of AFL game results it is reasonably straightforward to calculate a probability for any winning margin to occur. These probabilities seem to correspond reasonably well to reality. In my next article I will describe how the normal distibution model for AFL game results can be applied to determine probabilities during the course of a game (e.g. for "in the run betting")

Scott McIntosh runs the website Online Poker Room Reviews

Proceed to Part 3 of this series