This is a puzzle I presented at the MathsJam conference. It’s a problem that gave me a headache for a week or so, and I thought others might enjoy it, too. I do know the answer, but I’m not going to give it away — you can tweet me @icecolbeveridge if you want to discuss your theories! (As Colin Wright says: don’t tell people the answer).
You’ve heard of the Monty Hall Problem, right?
Well, just in case: the traditional presentation is that, at the end of an episode of the gameshow Let’s Make A Deal, the host – Monty Hall – would offer you a choice of three doors. Randomly arranged behind the doors are two goats, which you don’t really want; behind the third is a shiny car (which you do want). One prize per door, naturally.
You pick a door, and – rather than show you what’s behind it – Monty opens one of the other two doors to reveal a goat, before offering you a simple choice: stick with your original choice, or switch to the other closed door?
It’s a famous puzzle: when Marilyn vos Savant wrote about it, she reportedly received hate mail from irate mathematicians disagreeing with her (correct) solution; it’s also said that Erdős didn’t believe the answer until someone showed him a simulation.
The answer is, the first time you hear it, surprising: to maximise your chances of winning the car, you should switch doors. Switching is the correct move if you were originally wrong, and you’re more likely to have been wrong than right, so you should switch.
I don’t want to give you that!
Imagine you’re playing Who Wants To Be A Millionaire and a question comes up where – as far as you’re concerned – all four answers are equally likely. You chat with Chris about this, suggest a guess… and Chris then reminds you you still have a lifeline. You can go 50-50.
“Computer,” says Chris, “remove two random wrong answers!”
The computer makes a dramatic noise and removes two of the other answers, leaving you with a choice: do you stick with your original answer, or do you switch to the other?
You see: after you’ve made a choice at random, Monty Hall leaves you in a situation where you have one right and one wrong answer to choose from; Chris Tarrant leaves you in the same situation – so it would seem that switching to the other answer should give you a 75% chance of winning.
But, on the other hand, you didn’t go 25-75, you went 50-50 – and it seems absurd that making a guess before using your lifeline should affect the probabilities at all.
So, the question: which interpretation is correct? And how come?