Penalties and Bayes' Theorem

Real Madrid are on course to win La Liga title and there is a large majority of people that have been bemoaning the amount of penalties that Real have been awarded as the title race has hotted up.

To what extent is this fierce controversy true? Bayes' Theorem to the Rescue

Bayes' Theorem helps us update a belief based on new evidence by creating a new belief. In this case, it can help us update our belief of the probability of Real Madrid winning a game given that it is awarded with at least one penalty during the same game.

Descriptive stats provided by La Liga

So far, Real Madrid has:

• PLAYED 35 games
• WON 24 games (.69)
• NOT WON 11 games (losses and draws) (.31)
• WON 6 games with at least one PENALTY in favor (.25)
• WON 18 games with NO penalty in favor (.75)
• NOT WON 2 games with at least one PENALTY in favor (.18)
• NOT WON 9 games with NO PENALTY in favor (.82)

With these stats, we are ready to reach our goal:

We want to know the probability of Real Madrid winning a game given that it is awarded with at least one penalty during the same game.

Great. Now that we know our goal, we can move forward and begin with the analysis by using a decision tree.

Now, let's compute the value for each pathway.

• Path 1: Won Penalty = .69 x .25 = .1725
• Path 2: Won No Penalty = .69 x .75 = .5175
• Path 3: Not Won Penalty = .31 x .18 = .0558
• Path 4: Not Won No Penalty = .31 x .82 = .2542

Next, we need to sum Path 1 and Path 3 because this value will give us the probability of Real Madrid being awarded with at least one penalty during a game. This value is .2283.

Now, we will divide Path 1 by the value we arrived at in the previous step. This end result will be our updated probability. This gives us .7555, which is ~ 76%.

Conclusion

• The probability that Real Madrid wins a game is 69%.
• The probability that Real Madrid wins a game given that it is awarded with at least one penalty during the same game is 76%. Is this result the consequence of fair referees' choices? Feel free to update your own beliefs...

Learning Recommendations

• Bayes Theorem: A Visual Introduction For Beginners
• Understanding Bayes' Theorem