I was wandering around YouTube one weekend looking for something mentally stimulating when I came across the Monty Hall problem. For those who are not familiar, the problem is named after the host of the gameshow Let’s Make A Deal. On the show, there was a game where the contestant would be provided 3 doors; One had a car behind it, the others had goats. The contestant would choose the door they think had the car behind it, and then Monty would help them out by revealing one of the goats. The contestant would then be given the chance to stay with their first choice, or switch to the other unopened door. So what would you choose, and does it even matter? I generated some data to simulate it, and the results will probably surprise you.
Most people’s intuition will tell them that it doesn’t matter which they choose (the odds are even), but the above chart shows otherwise. Out of 10,000 generated attempts, a third of them win if they stay, and the rest win if they switch. Now that you’ve seen this simple chart, which would you choose? The choice seems pretty obvious given the data in front of us. Without this chart, most people would not even second guess their incorrect assumption.
What is interesting is that almost everyone gets this problem wrong, including many people with PhDs in Mathematics when this problem was first proposed (there are currently over 40 papers written on this problem and its variants). There are different explanations for this counter-intuitive phenomena that you can find all over the web if you look up “Monty Hall Problem,” so I won’t get too much into that. The chart is all you need to draw the correct conclusion.
Most people know enough about probability to handle basic odds. Flipping a coin and getting heads will happen half of the time, and similarly for rolling a specific number on various dice. In the same respect, picking a car from behind a series of doors is one out of however many doors there are. If I had asked you what are the odds a given door has the car behind it, you could easily tell me the answer.
When randomly picking a door out of 3 (like in the Monty Hall problem) it is a one third chance of winning. If randomly picking a door out of 2, it is a one half chance. If you flipped a coin to decide whether you stayed or switched after Monty revealed a goat, the even odds that most people expect would be true. So it isn’t that people’s intuition is wrong, it is just answering the wrong question!
The fundamental issue is that most people are assuming the doors are always independent like a coin toss. The doors are closer to a deck of cards though. Knowing the order of cards doesn’t change the odds of a random pick, especially if you only find out after you drew your card.
Now let’s say that you had some strange business that relied on people picking cars instead of goats. Average value of a goat is about $500, and a new car is about $30,000. The chart above shows the difference in average value between the 3 options. Staying is clearly a poor choice, as it is almost half of what always switching would bring. The intuitive choice of just letting people decide for themselves (since it doesn’t matter) is better, but still only about 3/4 of what you would get with switching.
So how much can you trust your intuition? When your personal reputation, money, company, and employee’s well being is on the line, can you afford to not make the best choice? Can you afford to not reassure your peers and colleagues that your decisions are sound? The right information can be worth a car.
Or at least a goat.