I played the Monty Hall game 1,000 times on montyhallsim.vercel.app, staying with my first door every time, and won 35% of them. I had expected something near 50, so I was already drafting my objection. Then I changed one setting to always switch, played another 1,000, and won 66.5%.
The site explained that switching works because it “capitalizes on the host’s knowledge.” I read that line about five times. It’s a perfectly good sentence and it did nothing for me.
I first met this problem in the end-term paper for Statistics for Economics at Ashoka University, Professor Ashwini Deshpande’s course. It’s named after Monty Hall, who hosted Let’s Make a Deal, and the paper gave me the usual setup. Three doors, one car, two goats, and a host who knows where the car is and never opens that door. Should you switch?
I had never seen it before, and that didn’t worry me. It looked like a clean little probability puzzle, the kind I enjoy, and I settled in, happy to have a proper problem.
The question was worth 10 marks out of 50. I spent about 30 of the 90 minutes on it, a third of the paper, which was not my best time management. There was a clock on the wall behind the invigilator’s desk. Every few minutes I looked up at it, did the sum on what this was costing me, felt a small pang about the other questions, and went back to the page. I’d decided I could solve it properly and I wasn’t going to write down a guess.
Say I pick Door 1 and the host opens Door 3. Bayes’ theorem gives the chance the car is behind each door given what I’ve just seen, so I gave each door a prior of one third and wrote out how likely the host was to open Door 3 in each case. I treated him as equally likely to open it whether the car was behind Door 1 or Door 2. Everything cancelled and I got a half for each door.
I can still see the page. It was on the right-hand side of the answer booklet, about two thirds of the way down, my handwriting getting smaller as the proof ran out of room. The last line said “Hence 50-50, no advantage in switching.” I underlined it twice and drew a box around it. Then I sat back, very pleased with myself. It matched my gut, which made me trust the algebra more.
I handed in the paper convinced those thirty minutes had been well spent.
Outside the hall there was one of those people who start dissecting the paper the second it ends, going question by question with anyone who stands still long enough. I was never one of them. I leave exams wanting to talk about anything else. He got to me anyway, and to the Monty Hall question, and said the answer was obviously to switch. He said it casually, as if everybody knew.
I stood there running my proof back through my head, line by line, and I couldn’t find a single wrong step. I knew I was wrong anyway. I felt stupid in a very specific way, the kind where you can name the thirty minutes you wasted. I skipped the group heading out for chai and walked back to my room alone.
Later I was shown where it broke. If the car is behind Door 2, the host has to open Door 3, because he has no other choice, so my “equally likely” was wrong. I can follow that sentence. I’d be lying if I said it settled anything.
Ten marks were gone, and a quarter of my time with them, so the rest of the paper got whatever was left. I bombed it. I want to put that on record. The question was fair and the mistake was mine. I was embarrassed for a week and irritated for years (till even today if I am being very honest, and as I write this, I feel even more irritated).
I can’t tell you what the other questions on that paper were. This one I could nearly redraw, from the Door 3 in my first line to the box at the bottom. I walked out proud of it. I had used Bayes’ theorem, the same theorem that gives the right answer, and a stranger undid all of it in about ten seconds. It didn’t end there, because I was shown the error, could follow it, and still looked at two doors and saw fifty-fifty. Seven years on, it hasn’t closed.
Two Doors, Obviously Fifty Percent
Here’s my case, and I still think it’s decent. I’ve rehearsed it enough times, mostly alone. The host opens a door and two are left. One has a car, one has a goat, and I’m standing at one of them, so it’s either right or wrong. If you’re reading this and also thinking two doors means fifty-fifty, hello. You don’t need a statistics course for it, which is a strange thing to say about a statistics exam.
The door doesn’t know it was chosen. The goat doesn’t know it was spared. I can’t see how removing an option makes my door worse than the other one.
That’s what I wrote in 2019, and it’s what I think every time I look at the screen. The answer key said to switch, since switching wins two-thirds of the time.
Believing the Number
While the counter climbed toward 1,000 on the switching run, I think I was hoping it would stall around 50. It didn’t. Two thousand rounds is a lot of rounds, I can’t argue with it, and I still sat there going, but there are two doors.
The hundred-door version I get. I pick one door out of a hundred, the host opens 98 goats, and obviously I switch, because he walked past 98 doors and left exactly one shut. My door is just the one I touched first.
What I can’t do is shrink that thought back down to three doors. People tell me the same logic applies, since my first pick had a one-in-three chance that never changed, which leaves the other door with the remaining two-thirds. I can say that sentence out loud. The simulator backs it up. Then I look at the screen and there is a door with a car and a door with a goat, and I’m back in the exam hall.
Whenever Monty Hall comes up at a dinner, I get quiet and start looking for the water jug. On the exam I wrote 50-50. The simulator wrote 66.5. I still can’t get the two of us to agree.
