The Monty Hall Problem: The Probability Puzzle That Divided Mathematicians

Nearly 1,000 PhDs Wrote Letters Saying She Was Wrong — She Was Right

In September 1990, Marilyn vos Savant published a reader-submitted puzzle in her Parade magazine column "Ask Marilyn" and gave an answer that triggered one of the most extraordinary public mathematical controversies of the 20th century. The puzzle was simple: you are on a game show, three doors stand before you, a car is behind one and goats behind the other two. You pick door 1. The host — who knows what is behind every door — opens door 3, revealing a goat. He asks: do you want to switch to door 2? Vos Savant's answer was yes, always switch. The switch wins two-thirds of the time. Approximately 10,000 readers wrote in to disagree, including nearly 1,000 with PhDs. They were wrong. The puzzle is now known as the Monty Hall Problem, named for the host of "Let's Make a Deal."

Why Intuition Fails

The near-universal wrong intuition is that after the host opens a door, two doors remain and the car is equally likely to be behind either one — a 50-50 proposition. This reasoning misses a critical structural feature: the host's action is not random. The host always reveals a goat, always from a door you did not choose, and always knows where the car is. The host's knowledge changes the information content of the reveal.

Probability depends on what is known. The host is not choosing randomly from the remaining two doors. He is constrained to reveal a goat and constrained to not reveal your door. This asymmetry is why the doors are not symmetric in probability after the reveal.

The Probability Analysis

There are three equally likely initial scenarios when you choose door 1:

Staying wins in 1 out of 3 scenarios. Switching wins in 2 out of 3 scenarios. The probability of winning by switching is exactly 2/3. Staying gives exactly 1/3. These are not approximations — they are exact values under the stated conditions.

The Bayesian Interpretation

Bayes' theorem formalizes the same result. Let C₁, C₂, C₃ denote the event that the car is behind door 1, 2, or 3. You chose door 1. The host opens door 3 (event H₃). What is the probability the car is behind door 2 given H₃?

• P(C₁) = P(C₂) = P(C₃) = 1/3 (equal prior probability)
• P(H₃ | C₁) = 1/2 (host can open door 2 or 3 with equal probability)
• P(H₃ | C₂) = 1 (host must open door 3 — the only remaining goat door)
• P(H₃ | C₃) = 0 (host cannot open door 3 — the car is there)
• By Bayes' theorem: P(C₂ | H₃) = [P(H₃ | C₂) × P(C₂)] / P(H₃) = (1 × 1/3) / (1/2) = 2/3

The posterior probability that the car is behind door 2, given the host opened door 3, is exactly 2/3. The car is behind door 1 with posterior probability 1/3. The math matches the table above.

Simulation Confirms the Math

Computer simulations of the Monty Hall problem consistently confirm the 2/3 result. In a simulation of 100,000 trials conducted by statistician at MIT in 2002, staying strategies won 33,337 times (33.3%) and switching strategies won 66,663 times (66.7%) — converging precisely on the theoretical values. Stanford mathematician Persi Diaconis performed similar simulations and confirmed identical results. The controversy was mathematical, not empirical.

Variants That Change the Answer

The 2/3 result depends on specific conditions. Changing those conditions changes the answer.

The "Monty Fall" variant — where the host randomly opens a door that happens to show a goat — results in 50-50 odds. The difference is whether the host's choice is constrained by knowledge. When the host acts randomly and a goat happens to appear, no information about the car's location is transmitted by the choice of door. The host's knowledge is the source of the asymmetry in the original problem.

Why Experts Got It Wrong

The academic letters Marilyn vos Savant received came from mathematicians, statisticians, and scientists. Several factors explain the widespread error. First, the problem is stated in natural language, which obscures the crucial detail that the host is not acting randomly. Second, human intuition consistently underweights conditional information — a phenomenon demonstrated in dozens of psychology studies on base rate neglect and the representativeness heuristic. Third, the problem superficially resembles a two-event uniform probability problem, which would yield 50-50. Paul Erdős, one of the most prolific mathematicians of the 20th century, initially gave the wrong answer before simulation convinced him. Elegant in its simplicity, devastating to intuition.