The Monty Hall problem has been explained a thousand times — so I’m not going to re-do the probability. Other people did a great job of that. What I want to talk about is a piece of history I ran into, and the thing it made me want to do: build the answer instead of arguing about it.
The part of the story nobody tells you
When I researched this, I found something I didn’t understand. There was a mathematician named Paul Erdős — I’m going to butcher the name — one of the greats. And the only way he actually believed the Monty Hall answer wasn’t the probability math, all those “if you do this, what’s your chance here or there” calculations. It was a computer simulation.
And what occurred to me is: we can replicate that today, pretty easily. Better than that — I can arm you to replicate it, because you can build the whole thing in Excel with a few simple formulas. So that’s the plan: let’s build a computer sim and watch it land on the answer.
The problem, quickly
There’s you. There’s Monty, the game-show host — and I always picture Monty with crazy hair. And there are three doors: 1, 2, 3. Behind one is a car; behind the other two are goats. Monty tells you to pick a door — you don’t know what’s behind any of them. Say you pick door 1.
Then Monty shows you a door, and he will always show you a goat. That’s the rule — the most important way to think about this. Now the whole crux of the problem: do you stay, or do you switch? The theoretical answer is that if you switch, you win about 2/3 of the time — 66.7%. But I don’t want to get into how that’s calculated, because I don’t think that’s the interesting part. The interesting part is: if we build a computer sim, we can see how close we get to that number.
Building it in Excel (you can copy this at home)
Here’s the whole thing, column by column, and anyone can rebuild it — take a screenshot of the formulas if you want:
- Trial — I run 1,000 of them.
- Car —
RANDBETWEEN(1,3): which door the car is behind. - Pick —
RANDBETWEEN(1,3): your guess. You don’t know the car. - The host’s move — this is the clever bit: I have to simulate the knowledge the host has. He never shows your pick and never shows the car — so I take the doors 1–3, remove your pick, remove the car, and whatever goat is left over is what he opens.
- Switch-to — the door you’d move to.
- Stay wins / Switch wins — just a 0 or 1: if the car equals your pick, staying wins; if it equals your switch door, switching wins.
Then the beauty of Excel: you grab the corner and drag the formulas down 1,000 rows — and that’s it, you’ve run a computer simulation. At the bottom I AVERAGE the 0s and 1s, times 100, and it spits out a clean percentage. My run came out to about 65% switch / 35% stay — right up against the theoretical 66.7%. For anyone who wondered what a “computer simulation” even is: we just ran a very simple one.
Run it yourself (no Excel needed)
Not everyone has Excel — so I built a version you can just click. Run 1,000, run 100,000, run a million. Watch what happens: the more trials you add, the closer it hugs the theoretical value, and the less it jumps around. That’s the whole idea — you’re approaching the limit, and the limit is 2/3.
[ Monty Hall simulator embeds here — see the note above. ]
Prefer the spreadsheet, download it here:
Why I actually did this
Here’s the real point. Say you don’t understand the Monty Hall problem. You can take the rules of it, model it — like we did, in a simple Excel document — and then ask yourself, “Do I actually get what these people are saying?” That’s the beauty of having a critical mind.
And I want you to think about it this way: sometimes it’s much easier to find a contradiction than a proof. Let that settle in. Because if you find one contradiction, it invalidates the whole assumption — and in math and engineering, that’s one of the most powerful things you can do. By running a computer sim, we set ourselves up to find contradictions: just model the behavior and see what happens.
This is a MavonX Side Quest — same desk, no sudoku, just a claim worth testing. I make original, never-repeated sudoku (going for a Guinness World Record). Free daily puzzles and the books are at mavonx.com. Don’t trust it — test it.