STAGE 1: THE BASIC MODEL

Most and sports web sites and newspapers print statistics for every domestic game, so Monday mornings during the football season offer the chance for some relevant data collection. As well as match scores, I recorded the number of shots on target for each team in 151 English league games. Your students can do this for themselves, but my games produced an average of 8.22 per team, resulting in the average number of goals being scored as 1.43 . (Which in turn means that you need on average 5.74 shots to score a goal.)

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These figures struck me as being almost magically convenient - for two of us to play a match, all we need to do is:

each of us rolls an ordinary die eight times, one for each scoring chance. Every time you roll a 6 you score a goal.

Games don't come in much more modest formats than this, and children will get lots of fun out of this simplest of all mechanisms. Don't waste the opportunity of keeping records of all the games the pupils play. By my calculations we can expect the following distribution patterns of goals scored:

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0 goals - 23%

1 goal - 37%

2 goals - 26%

3 goals - 10%

4 goals - 3%

5 goals - 0%

 

e.g. in every 100 matches a team is likely to score just one goal 37 times. These figures seem pretty realistic, which is a nice sign.

How realistic are these figures? Let's return to the statistics page of the newspaper. I checked the entire season's results for the Premiership, and was delighted to find that the actual scores were remarkably close to the above distribution:

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0 goals - 32%

1 goal - 32%

2 goals - 21%

3 goals - 9%

4 goals - 4%

5 goals - 1%

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STAGE 2: ADDING A LITTLE FLAVOUR

I said that a simulation can allow you to build in extra features on top of the basic idea. For example, some of a game's most exciting moments are when an attempt doesn't directly result in a goal but leads to a second attempt. Perhaps the initial shot hits the crossbar and rebounds into play, or the goalkeeper saves but the ball runs clear for an attacker to have a second chance. So to our basic rule (6=goal) we'll add a second: if the number rolled was a 5 then a follow-up roll is made. For the follow-up the same rules apply (6=goal, 5=yet another attempt).

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Of course, this is going to increase the attackers' success rate and the number of goals scored, so perhaps we ought to allow the defending side some comparable advantage. Let's go back to the original rule and modify it slightly: when a 6 is thrown the defending side rolls, and if a 1 results then the "certain" goal doesn't materialise after all - the goalkeeper makes a miracle save, or a defender manages to block the shot, or perhaps the referee rules an attacker offside and disallows the goal.

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So the basic structure now looks like this:

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Both sides get eight goal attempts. For each attempt the attacker rolls: a 6 indicates a likely goal (though the defender gets a last-ditch saving roll for which a 1 is needed). An attempt which generates a 5 merits a further attacking roll, for which the normal rules apply.