How You Can Win a Sportscar, or: On the Importance of Priors

I run in the morning a few times a week. My older dog and I need the workout (the younger one isn’t old enough yet), and depending on my mood it’s either a mindfulness meditation session, or I put my thinking cap on and ponder how to solve the problems of the world.

This morning it was the latter, and the question I pondered was "why do so many people not want to win a sportscar"? The more geeky of our readers (i.e. all two of you?) will already have guessed that I am talking about the brain teaser “The Monty Hall Problem” (MHP), and the fact that so many people have a problem with understanding the correct solution.

I won’t go into the history of the puzzle itself here – the origin is likely much older than the Monty Hall Show “Let’s Make a Deal” but since it’s been popularized under that name, that’s what stuck. It goes like this:

Imagine you are on a game show and are shown three doors. You are told that behind one door is a sports car, and the other two goats. You are then asked to chose a door and told you will win what’s behind it. [1]

The twist comes after you chose a door. The host now opens a door that a. you didn’t choose and b. doesn’t have the sports car behind it. (These two constraints are important.) She asks whether you want to stick by your first choice or switch to the other door.

What do you do? And what are your probabilities?

The overwhelming majority of people when posed the MHP say it doesn’t matter. Two doors left, car behind one of them, the chances are 50-50, right? Wrong; the chances are 2/3 - 1/3 for switching!

Why? This gets us right into the heart of our discussions on risk assessments and why so many people are bad at it. The key concept here is so-called “priors”.

Priors, or prior probability distributions, are a concept of Bayesian statistics (a topic for another post / post series). Simply put, it is additional information that affects the (subjective) probability distribution of a variable.

Take, for example, a 6-sided die. Normally, we would all go with the so-called “frequentist” approach and assume that the probability of rolling a six is p=1/6. (I am lazy, so going forward I will write p when I mean probability.) Leaving aside some of the more superstitious gamblers who believe in streaks, everybody nowadays understands this. But what if you had done an experiment with that exact die, and after 60,000 rolls, the six only turned up 9,000 times, not the 10k you’d expect? [2] You’d assume the die was loaded – your prior has changed.

When you start out the MHP, you have no priors – p=1/3.  After the game master opens another door, your priors change. Why? Because of the two rules that she will neither open your door, nor a door with the sports car behind it, i.e. her choice isn’t random. To illustrate, let’s assume for a moment that her choice actually was random. The decision chart would look like this [3]:

Each of the 6 options has a probability of 1/6. With this rule set, you win twice by switching, win twice by not switching, and twice it’s game over because the GM uncovered the car. I.e. 50-50 is correct.

But with the rule that she can’t open the car door, the chart looks different:

All variations still have p=1/6, but e.g. Variation 3 can’t happen anymore – door 1 is still off limits (your initial door) but door 2 now is, too. Variation 4 now “inherits” V3, and has a total p=2/6. Same with 5 and 6. You know more than you did in the random variation, so you win 4 out of 6 times by switching.

If you still don’t believe it, consider the following rule change: there are 1,000 doors. You pick one (p=1/1,000 of being the right one). Now the GM opens all doors except one, with the same two rules that she can’t open your door and can’t open the car door. Same game as with three doors, but the GM now eliminated a huge degree of uncertainty for you. Still believe the probability is 50-50 now?

The lesson for risk strategies: priors are important. Risks seldom exist in a vacuum, and good risk takers consider as much extra information as possible.

Notes:

[1] I don’t see what’s wrong with winning a goat – we are considering getting some for the milk and cheese and to save on trimming back brambles and stuff – but I guess the standard assumption is that a sports car is the preferable win

[2] If you are interested in the probability of that happening, ask for the formula in the comments. Excel can’t deal with factorials this high – suffice it to say that p is pretty low.

[3] For simplicity’s sake I am just showing you initially chosing door 1 in both charts. Chosing other doors doesn’t change anything as the problem is symmetrical.