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# Gambling related articles

### Introduction

By Scott McIntosh

"In the run betting" is a popular betting option offered by bookmakers. In the third and final part of this series of articles I will describe how the normal distribution model of AFL game results can be applied to calculate probabilities during the course of a game.

### Method

Given the amount of time remaining and the current score difference in a game we can calculate probabilities of the game result based on the before the game predicted line and standard deviation. These calculations assume that the predicted line and standard deviation per time unit does not vary during the course of a game.

Given:

• line = before the game predicted line
• stdev = before the game standard deviation
• time = proprotion of game remaining (e.g. 1 = before game, 0.75 = quarter time, 0.5 = half time, 0 = game over)
• score = current score difference

We calculate:

• Adjusted Line = (line * time) + score
• Adjusted Standard Deviation = stdev * square root(time)

Using the adjusted predicted line and standard deviation the probability of any result occuring can be calculated as described in Part 2.

Example

Team 1 is a 15.5 point favourite and is behind by 20 points at quarter time. We want to calculate the probability and fair odds for each team winning the game.

Adjusted Line = (15.5 * 0.75) + (-20) = -8.375
Adjusted Standard Deviation = 38 * square root(0.75) = 32.98097

Using Excel as described in Part 2
Probability Team 1 wins = normdist(999, -8.375, 32.98097, true) - normdist(0.5, -8.375, 32.98097, true) = 0.393702
Fair Odds = 1 / 0.393702 = \$2.54

Repeating this procedure for all results we obtain the table below.

Winner Before Game Quarter Time
Probability Fair Odds Probability Fair Odds
Team 1 0.653481 \$1.53 0.393702 \$2.54
Team 2 0.336858 \$2.97 0.594562 \$1.68
Draw 0.009660 \$103.52 0.011736 \$85.21

### Accuracy

Graph 3 represents how well the normal distribution model fits time periods smaller than the whole game, a major assumption for calculating in the run probabilities. The result (i.e. score difference) for the second half only was compared to what the model predicted. The before the game predicted line used is the bookmakers line and the second half adjusted line is 0.5 times the bookmakers line. The standard deviation used for the predicted frequency curve is 26.87006 (38 * square root(0.5)). From observation there is a close fit between the actual and predicted results which statistical tests confirm.

### Graph 3: Second Half Results (1056 Home and Away Games : 2000 - 2005) ### Limitations

The major assumption for calculating in the run probabilities is that the predicted line and standard deviation do not change over time. Whilst this may be true for most of the game towards the end of the game this may not be true. A team winning by a small margin may "waste time" to increase their winning chances. This would have the effect of decreasing the standard deviation and average winning margin.

Another limitation towards the end of the game is the discrete scoring nature of the game, a goal is worth six points and a behind 1 point. With time remaining for 1 scoring shot only there is no difference if a team is winning by 2, 3, 4 or 5 points although the model would say there are significant differences.

### Conclusion

Despite the relative simplicity of the normal distribution model and the assumptions that are made, reasonably accurate estimates of probabilities can be made before and during the course of an AFL game. More accurate predictions might be able to be made using more factors (e.g weather) which would require more data and further investigation.

Scott McIntosh runs the website Online Poker Room Reviews