Monty Hall Problem Calculator — Stay vs Switch Probability
Enter the number of doors and how many the host reveals, and see the exact probability of winning by staying or switching — and by how much switching beats staying.
Switching wins 2× more often than staying
- 1
P(win | stay)
1 ÷ 3 = 33.33 % - 2
Remaining doors after reveal
3 − 1 − 1 = 1 - 3
P(win | switch)
(3 − 1) ÷ (3 × 1) × 100 = 66.67Switching concentrates the remaining probability onto fewer doors.
How does this calculator work?
Classic Monty Hall (3 doors, 1 revealed): P(stay) = 1/3 ≈ 33.3%, P(switch) = 2/3 ≈ 66.7% — switching wins twice as often. Generalised (N doors, K revealed): P(stay) = 1/N, P(switch) = (N−1)/[N×(N−K−1)]. The advantage of switching grows as N increases or more doors are revealed.
Formula
How this is calculated
In the classic Monty Hall problem there are 3 doors: one hides a prize, two hide goats. You pick a door. The host — who knows what is behind every door — opens one of the other doors to reveal a goat, then asks if you want to switch. Switching wins with probability 2/3; staying wins with probability 1/3.
The intuition is that your initial pick has only a 1-in-3 chance of being right. The remaining 2/3 probability that the prize is behind one of the other two doors gets concentrated onto the single unopened door after the host reveals a goat, because the host cannot reveal the prize or your door. Staying keeps the original 1/3 chance; switching captures the transferred 2/3.
The generalised formula for N doors with the host revealing K empty doors is: P(win | stay) = 1/N and P(win | switch) = (N − 1) / [N × (N − K − 1)]. When K = N − 2 (the host opens every possible empty door), switching wins with probability (N−1)/N — dramatically better than staying at 1/N, and approaching certainty as N grows. The host must always open only empty doors and never your original door.
Frequently asked questions
Your initial pick is wrong 2 out of 3 times. When you are wrong (2/3 of the time) and switch, you always land on the prize — the host has eliminated all other wrong doors. When you are right (1/3) and switch, you land on a goat. So switching converts your 2/3 "wrong picks" into wins.
Yes — this is the crucial rule. A host who randomly opens a door (and might accidentally reveal the prize) produces different probabilities. The classical Monty Hall result requires the host to always knowingly reveal an empty door.
P(win | stay) = 1/100 = 1%. P(win | switch) = 99/100 = 99%. You should almost certainly switch — the host has essentially pointed you at the prize.
Also known as
TG we-Calculate Editorial Team. (2026). Monty Hall Problem Calculator — Stay vs Switch Probability [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/monty-hall-problem-calculator
TG we-Calculate Editorial Team. "Monty Hall Problem Calculator — Stay vs Switch Probability." TG we-Calculate. 2026. https://we-calculate.com/calculator/monty-hall-problem-calculator.
TG we-Calculate Editorial Team, "Monty Hall Problem Calculator — Stay vs Switch Probability," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/monty-hall-problem-calculator
@misc{wecalculate_monty_hall_problem_calculator, title = {Monty Hall Problem Calculator — Stay vs Switch Probability}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/monty-hall-problem-calculator}}, year = {2026}, note = {TG we-Calculate} }
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