The Two Door Riddle

My kids liked this puzzle from the movie Labyrinth:

Let’s imagine the following scenario.

There are two genotypes of people who encounter the two door riddle. Genotype A has an 80% chance of solving the puzzle, asking the right kind of question, and choosing the correct door (thereby avoiding certain death). Genotype B has a 20% chance of solving the puzzle and choosing the correct door. However, both Genotype A and Genotype B will guess randomly between doors if they cannot solve the puzzle, so they still have a 50% chance of choosing the correct door and avoiding certain death, even if they don’t solve the puzzle.

So these are the outcomes observed:

% Who Avoid Certain Death
Genotype A 80% (Who Solve Puzzle) + 50%*20% (Who Don’t Solve the Puzzle)=90%
Genotype B 20% (Who Solve Puzzle) + 50%*80% (Who Don’t Solve the Puzzle)=60%

If the two genotypes are each 50% of the population, and if you had 100,000 people encountering the two doors, the cross-tabs would look roughly like this:

Genotype A Genotype B Total
Chooses the Correct Door 45,000 30,000 75,000
Certain Death 5,000 20,000 25,000
Total 50,000 50,000 100,000

As best as I can calculate it, the heritability of solving the puzzle is 0.36, and the heritability of avoiding certain death is only 0.12. (Or, the correlation of solving the puzzle with genotype is 0.6 and the correlation of choosing the correct door and avoiding certain death with genotype is about 0.35.)

Two thoughts about this:

a) What causes someone to survive the Two Doors Puzzle? Is it their genotype, their brain, how well their brain works right then, their luck, asking the right question, stepping through the correct door?

All of them and none, right?

Defining “cause” solely as a hypothetical treatment in an experiment, as social scientists often would like to do, sometimes leaves out a lot of what we mean by causes.

b) The example above was written to indicate a huge role for genes and a moderate role for luck, but for the outcome we really care about (surviving), the heritability was only 0.12. It’s pretty remarkable that, for example, real-life estimates for the heritability of income (which must involve a fair amount of luck) are considerably higher than this, and for more strictly biological traits estimates are higher still.

One thought on “The Two Door Riddle

  1. Your point (which is an excellent one) is that even traits with a strong advantage don’t always get passed on all that often.

    However, consider also the fact that Darwin favors traits that lead to more offspring. So if people who pick the wrong door also tend to sleep around more (recklessness or whatever), their genes will be passed on even if they’re objectively worse for their offspring. Which, I think, explains why the mean IQ isn’t 150.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s