[Math] How to calculate probabilities of a football team scoring a certain number of goals

probabilityprobability distributions

If a team has 50-50% chances of scoring over or under 1.5 points, how do you calculate what are their chances of scoring exactly 0,1,2 or 3 points? And how do you calculate the chances of scoring over/under 2.5 points? (The team can only score 0,1,2,3,4 and so on, no decimals)

Later edit: The chances for team to score 1.5 points comes from a match were the expected total number of goals is 2.5 (50%-50% chances of both teams to score over or under 2.5) and team A is better than team B with 0.5 points. If this helps.

Best Answer

If all you know is that the chances of scoring under or over $1.5$ points, then there is no way to know what the chance of scoring exactly $0$ points is.

For example, the two cases:

  • Chance of scoring $0$ is $0.5$, chance of scoring $1$ is $0$, chance of scoring $2$ is $0.5$
  • Chance of scoring $0$ is $0.25$, chance of scoring $1$ is $0.25$, chance of scoring $2$ is $0$, chance of scoring $10$ is $0.5$

both result in the team having a 50-50 chance of scoring over or under $1.5$ points.

Basically, you want to calculate $p_0, p_1,p_2,p_3\dots$ while all you know is that

  1. $p_0+p_1=\frac12$
  2. $p_2+p_3+\cdots = \frac12$.
  3. $p_i\geq 0$ for all $i$.

There are infinitelly many solutions to the above set of equations.

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