Monty Hall Problem Answer: Switch Doors [Solved]

The Monty Hall problem answer: switch doors. When the host offers the swap, switching wins the car 2 times out of 3, while staying with your first pick wins only 1 time out of 3. Always switch.

This is the Monty Hall problem answer that surprises almost everyone the first time they hear it, and it is one of the most asked probability questions in finance and trading interviews. It regularly appears on lists of top quant interview questions alongside the classics. Three doors, one car, two goats, and a host who knows exactly where everything is. Here is the full walkthrough.

The Puzzle, Restated Simply

You are a contestant on a game show. There are three doors. Behind one door is a car; behind the other two are goats. You pick a door, say Door 1. The host, who knows what is behind every door, opens one of the other two doors, say Door 3, revealing a goat. He then asks: do you want to stay with Door 1, or switch to Door 2?

The critical rule, and the one most people gloss over: the host ALWAYS opens a door with a goat behind it. He never reveals the car. His choice is not random; it is constrained by his knowledge.

The Monty Hall Problem Answer, Step by Step

Walk through all three possible worlds, assuming you first pick Door 1:

World 1: the car is behind Door 1 (probability 1/3). You picked right. The host opens either Door 2 or Door 3 (both goats) and offers the swap. If you switch, you lose. If you stay, you win.

World 2: the car is behind Door 2 (probability 1/3). You picked wrong. The host cannot open Door 2 (that would reveal the car), so he must open Door 3. If you switch to Door 2, you win. If you stay, you lose.

World 3: the car is behind Door 3 (probability 1/3). You picked wrong again. The host must open Door 2, the only remaining goat. If you switch to Door 3, you win. If you stay, you lose.

Add it up: switching wins in Worlds 2 and 3, that is 2/3 of the time. Staying wins only in World 1, that is 1/3 of the time. The three worlds are equally likely, so the arithmetic is airtight.

Why Switching Wins 2/3 of the Time

The cleanest way to see it: your first pick is wrong 2 times out of 3. In every world where your first pick is wrong, the host is forced to point at the car. By opening the one remaining goat door, he funnels all that probability onto the unopened door. That concentration effect is the real Monty Hall problem answer: your 2/3 chance of picking wrong on the first guess becomes a 2/3 chance of winning when you switch.

If your intuition still resists, scale it up. Imagine 100 doors, one car, 99 goats. You pick Door 1. The host, who knows everything, opens 98 other doors, all goats, leaving just your Door 1 and Door 37. Do you stay? Of course not. Your first pick had a 1/100 chance; the other 99 doors collectively held 99/100, and the host has now concentrated all of that probability onto Door 37. Three doors work exactly the same way, with 2/3 instead of 99/100.

The Trap Most Candidates Fall Into

The classic wrong answer is “it is 50/50 now, so switching does not matter.” The reasoning goes: two doors are left, the car is behind one of them, so each must be 1/2. The error is treating the two remaining doors as symmetric. They are not. The host’s action was information-rich: he used his knowledge of where the car is to choose which door to open. Your Door 1 never received that treatment. Its probability stays at 1/3 while the other door absorbs the 2/3.

Sharp interviewers then twist the question with a follow-up: what if the host does NOT know where the car is and opens a door at random, and it happens to be a goat? Now the symmetry is real, and the answer changes: switching no longer helps, it genuinely is 50/50. The takeaway is that the 2/3 answer depends entirely on the host’s knowledge. State that assumption out loud in the interview and you separate yourself from the pack.

What the Interviewer Is Testing in the Monty Hall Problem

First, conditional probability. Can you update your beliefs when new information arrives? Finance runs on this skill: every market data point is a door the host just opened.

Second, assumption discipline. The whole solution hinges on the host knowing where the car is. Candidates who say “switch” without stating that assumption have memorized, not understood. Say the assumption first, then the answer.

Third, resistance to false intuition. “Two doors, so 50/50” feels right and is wrong. Trading desks are full of strategies that feel right and lose money; the interviewer wants people who trust the math over the gut.

Finally, communication. The best answers walk through the three worlds crisply, name the 2/3 number, and close with the host-knowledge caveat. That is the entire performance in about ninety seconds.

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