Every couple of years, I run into this probability puzzle that reminds me just how bad humans are at assessing probabilistic outcomes. Its called the Monty Hall problem, named after Monty Hall, the host of the famous gameshow, Lets Make a Deal.
The puzzle goes something like this.
Suppose you are given the opportunity to choose between three doors. Behind one door is a prize of significant value; say $1 million. Between the other two doors are near valueless prizes. Pick a door, any door.
Now that you have picked a door, the host, Monty, opens one of the two remaining doors, revealing that there is a pile of coal behind the door. The $1 million is clearly still behind one of the doors that remains closed; perhaps the door you originally selected. To make things interesting, Monty gives you the opportunity to switch doors; you can keep the door you selected originally, or switch to the other door that remains closed.
So, do you want to switch doors? Are you better off keeping the door you originally selected or switching to the other closed door. I know what you are thinking; it doesn’t matter right. There is a 50% chance the price is behind the door you picked and a 50% chance it is behind the other door. After all, there are only two doors remaining and you get to pick one. You might as well keep the door you originally picked.
What if I told you that your instincts are wrong and that the prize is two times more likely to be behind the other closed door than the door you originally selected. You’re thinking “Derek is crazy; this non-linear stuff has eroded his ability to think logically.”
Lets start over and try the exercise with 100 doors. You picked 1 of 100 doors; you have a 1 in 100 chance of picking the door with the prize on the first try. Now the game show host opens 98 of the other 99 doors, revealing that there is nothing behind them and leaving only the door you selected and 1 other door closed. Remember, you had a 1 in 100 chance of getting the door right on the first try. That means that there was a 99 in 100 chance the the prize was behind one of the other 99 doors. But 98 of them have been opened with nothing behind them.
Do you see now, there is a 99 in 100 chance that the other door contains the prize and only a 1 in 100 chance that the door you originally selected contains the prize. In the three door example, there is a 1 in three chance the prize is in the door you originally selected and a 2 in 3 chance the prize is behind the other door that remains closed.
Still don’t believe me, this Wikipedia post has a lengthy explanation here.
Our brains just aren’t very good at dealing with probabilities; or the randomness we face in the world. I get a kick out of that and try to keep that in mind whenever I’m making a big decision.