## July 24, 2006

### Soccer Probabilities

Although the USA could be talked into hosting the soccer world championship in 1994 and millions of people went to watch the matches soccer has never really become popular in the US. I still remember comments from American sports reporters stating that the game was simply boring with so few goals scored during the matches. They also proposed enlargening the goal so that more goals would be scored. But would that really make soccer more interesting?

Mathematicians tell us it would not - because more goals scored during a match would increase the chances of the better team to win even more. Now, how can this be? Consider two teams, A and B, with A being twice as good as B. This will translate into the probability of 2/3 for A and 1/3 for B scoring the next goal. Now let's see what happens if different numbers of goals are scored during a match:

• 0 goals B gets 1 point!

• 1 goal
The probability for A to score this goal is 66% (2/3) and 33% (1/3) for B.

• 2 goals
Probability for A to win 2:0 is 44%
Probability for B to win 2:0 is 11%
But the probability that the match will end in a draw is also 44%!

• 3 goals
Probability for A to win 3:0 is 30%.
Probability for A to win 2:1 is 44%.
That leads to an overall probability of A winning the match of 74% as a draw is not possible in this case.

• 4 goals
Probability for A to win 4:0 is 20%
Probability for A to win 3:1 is 40%.
Overall probability for A to win one way or other: 60%

• 5 goals
Probability for A to win 5:0 is 13%
Probability for A to win 4:1 is 33%
Probability for A to win 3:2 is 33%
Overall probability for A to win is 79%

• 7 goals
Probability for A to win 7:0 is 6%
Probability for A to win 6:1 is 20%
Probability for A to win 5:2 is 31%
Probability for A to win 4:3 is 31%
Overall probability for A to win is 88%
• These examples show that the probability for A to win the match increases with each goal that is scored in a match. What I found most interesting that if only two goals are scored in the match the probability to actually win for A only 44% and equal to the probability of the game ending in a draw, although the team is supposed to be twice as effective at scoring as their opponents.

Funny how calculations can sometimes prove your intuition wrong.
I think we should not tamper with the dimensions of the goal. Who would not like to see the underdog win against all odds?

More details of the calculations are given in the extended post. Have a try yourself!

As one sports reporter said during the last soccer worldcup: "The match ended in a draw, 1:1, but it could easily have been the other way round!"

Here's the example for 7 goals, the others work accordingly. A could win with the following results:

7:0 p=27/37
6:1 p=(7!/(1!(7-1)!)) x 26/37
5:2 p=(7!/(2!(7-2)!)) x 25/37
4:3 p=(7!/(3!(7-3)!)) x 24/37

The overall probability of A to win is then simply the sum of all.

Posted by Mausi at July 24, 2006 09:20 PM

http://mt3.mu.nu/mt/mt-tb.cgi/4519