P-value explained with Squid Game

Warning: *Spoilers for the TV show Squid Game Ahead*

I have seen countless articles explaining the P-value on medium and the internet. Here is the definition of p-value:

The P value, or calculated probability, is the probability of finding the observed, or more extreme, results when the null hypothesis (H0) of a study question is true

So I decided to take a stab at explaining p-value using my current favorite TV show Squid Game!

Now let’s set up our experiment!

The Squid Experiment

For those who do not know what Squid game is: Its a series of children’s games for hundreds of cash-strapped contestants and the winners get around $38 million USD cash prize. If you lose, well you die.

Now in the sixth game, the player Ali is faced with a dilemma. His partner Sang-Woo has offered to help him win round six of the game. He suggests a plan with which they can both win.

To trust, or not to trust?

Let’s say Ali will not believe Sang-woo if the probability of SangWoo helping a teammate is less than or equal to 0.5.

So Ali decides to jot down his data and create probability distribution of SangWoo helping his friends:

Now we create the normal distribution based on this data:

The distribution shows is less than 1% chance that SangWoo does not help friends according to Ali’s current data. Hence Ali is 99% confident that SangWoo will help his friends.

So now in Ali’s head here is the hypothesis from the current data:

H₀ : SangWoo wants to help Ali or P(SangWoo helping friends≥0.5)

H₁: SangWoo does not want to help Ali or P(SangWoo helping friends<0.5)

and Ali strongly believes his Null hypothesis is true since there is less than 1% chance that SangWoo will not help his friends. Our alpha becomes:

alpha = 0.01

Quick Note: Why do we do hypothesis testing?

Why do we do experiments or hypothesis testing?

When we do any experiment, we are essentially verifying if our null hypothesis (or our current belief) holds true. And by extension we are verifying if our null distribution still holds true. When we talk about 95% confidence in our data we are saying:

If we sample the data multiple times we are confident that the mean of the sample will be lie between \bar{x}\pm 2* \sigma if it is a two tailed test and
the mean of sample is greater than \bar{x}\>\pm1.6* \sigma in case of one tailed test (depending on left tail or right tail test)

Coming back to our problem, in our case it is a left tailed test, since out alternate hypothesis states the ‘<’ sign.

Note:

• If the alternate hypothesis had ‘>’ sign it would be a right tailed test.
• If the alternate hypothesis had ‘ \neq ’ sign then we will do a two-tailed test.

In Ali’s data, there is so little chance of SangWoo not helping his friends (approx 0.01%), that if it occurs that implies Ali’s distribution does not hold true and the alternate hypothesis is true.

Back to the Squid Game Experiment

Did Ali’s distribution hold true?

: Drum Roll :

Turned out Sang-Woo had no intention of helping his friend!

The moment Ali finds out the new observation

And if I put it mathematically:

P(SangWoo Helping his Friend) = 0.3

( The mathematical number 0.3 here implies Sang Woo has no intention of helping but he feels guilty. )

And what is the p-value for this observation in Ali’s distribution?

Coming back to the definition:

The P value, is the probability of finding the observed, or more extreme, results when the null hypothesis (H0) of a study question is true

Let’s break this down:

What is the null hypothesis? Ali’s current distribution holds true

Then p -value will be =

P(SangWoo Helping his friend ≤ 0.3 | current distribution is true)

= 0.0000045

p -value < 0.01 (alpha)

In distribution this value lies here:

This new observation proves that Ali’s null hypothesis needs to be rejected and the distribution which he created for SangWoo does not hold true.

Now Ali is “eliminated” and we need to come up with a new distribution for treacherous SangWoo.

Conclusion

In a nutshell p-value is the probability of an extreme case happening given your current distribution of population holds true (which is the null hypothesis). Another thing we learn here is it’s important to have high number of samples before jumping to conclusion. Just five games, really Ali?