How can we quantify ‘Luck’ and why can’t we be lucky all the time?

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My whole childhood I used to get very annoyed when people based the fact that they won or lost a match was due to ‘luck’.

  • “We lost the hockey game because the other team scored some lucky goals”
  • “We were lucky they didn’t score that last goal”

This annoyed me for the simple reason: I did not know how to quantify luck and I didn’t fully understand the science behind it.

Recently I have spent a lot of time researching luck and performance related to luck. Especially media stories which decribed the performance of a player or team as being lucky or unlucky. By breaking down what they are actually trying to describe, luck can be defined (in some sorts) by looking at the variance in our muscles when performing movements.

Written as an equation using the example of scoring a free throw in basketball:

Baskets scored = Skill + Luck

Skill = Time Practiced + Technique

Luck = Motor Control Variance

Breaking this down, if the player usually scores 4/10 baskets as an average when performing this exercise. How can you explain that some times this player scores 2/10 or even 8/10? Simply by quantifying luck: The players muscles perform in a different way (motor control variance) each time as it is virtually impossible to perform the EXACT movement in repetition even for elite athletes. Lets say that on the time that the player scored 8/10 their motor control happened to aid him in scoring more as their variance landed the ball into the basket more often, “Lucky”.

Luck and performance are often talked about as the same thing. One week a player can be “lucky” and the next they can be “average” or even “unlucky”. Using another sport as an example, lets compare two players playing golf in a competition:

Jack

Score on day 1:

8 under par

David

Score on day 1:

1 over par

Assuming that these players are the same skill level, it is safe to say that media coverage would say that Jack was very lucky that day and that David was neither lucky or unlucky. We now know that this is a cause of Motor Control Variance which was explained before. Now if you were asked to predict the score for day 2 based on the previous results most people would say that both Jack and David would score similar to day 1 but just by watching golf once or twice we can see that this is rarely the case. For example if Jack scored 4 under par on the 2nd day he would have “Been lucky on the first day and ran out of luck”. This can be worked out using math as a rough example:

  • The chance of Jack being very luck and scoring 8 under par = 5%
  • The chance of Jack being lucky and scoring 4 under par = 15%
  • The chance of Jack having average luck and scoring between -4 and +4 = 60%
  • The chance of Jack being unlucky and scoring 4 over par = 15%
  • The chance of Jack being very unlucky and scoring 8 over par = 5%

Although this is rough example it is apparent that it was highly unlikely (5%) that Jack scored 8 under par on day 1 and much more likely that he will score between -4 and +4. Which means that if Jack scores -3 on day 2 it is much more likely then scoring -8 again. The rule I have just spoken about is called “regression to the mean”

The Cambridge Dictionary of Statistics defines this as:

In statistics, Regression toward (or to) the mean is the phenomenon that if a variable is extreme on the first measurement, it will tend to be closer to the average on its second measurement – and if it is extreme on its second measurement, it will tend to the average on its first. 

Now you understand why many people make mistakes when analysisinp performance the following examples will make more sense to you:

When training pilots the flight instructor noticed that when someone performed in the simulator perfectly, they were then praised, and on the next try they performed worse. On the other side if a pilot performed extremely bad they were shouted at and the next time they performed better. Why?

Regression to the mean.

Most likely they would have performed worse after the perfect simulation regardless of what the instructor said.

A very obvious example is the “Sports Illustrated Curse”. It is often said that after an athlete features on the cover of the magazine the following year their performance is worse. It is actually quite simple to explain this: If a player features on the front cover of the magazine they have obviously been performing very well during the year, maybe you could say they have been extremely lucky. Now if we think about the regression to the mean it can help explain why they perform worse the next year. It is more likely that the 2nd year their performance will be closer to the average.

So next time your team has a very good/bad performance think about what luck actually is and how this will usually regress to the mean.

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