HesapHanem

Monty Hall Probability Calculator

Enter the number of doors to compare staying versus switching.

How it is calculated

The model has N doors, one prize, and a host who knows where it is. You pick a door, then the host opens exactly one losing door that you did not pick, and you may switch to any of the other N − 2 closed doors at random. Staying wins only if your first pick was right: P(stay) = 1 ÷ N. Switching wins when your first pick was wrong, which has probability (N − 1) ÷ N, and you then land on the prize with chance 1 ÷ (N − 2): P(switch) = (N − 1) ÷ (N × (N − 2)). The advantage is P(switch) ÷ P(stay) = (N − 1) ÷ (N − 2), shown with two decimals. N must be a whole number of 3 or more.

Worked example

Take 5 doors. If you stay, you win only when your first pick was the prize: 1 ÷ 5 = 20.00%. If you switch, your first pick must have been wrong, probability 4 ÷ 5, and then you choose one of the 3 remaining closed doors, so the chance is 4 ÷ (5 × 3) = 4 ÷ 15 = 26.67%. The advantage is 26.67 ÷ 20.00 = 4 ÷ 3 = 1.33×. The tool shows switch 26.67%, stay 20.00%, advantage 1.33×.

Try this example →

Common mistakes

  • Assuming the host opens doors at random. The result depends on the host knowing where the prize is and always opening a losing door; if the opened door could be the prize, the odds change.
  • Expecting the host to open all but one door. This tool models the host opening a single door, so with 10 doors you still pick among 8 closed ones, and switching is only 1.13× better.
  • Believing it is 50/50 after the reveal. The two remaining doors are not equally likely; your first pick keeps its original 1 in N chance.
  • Typing 2 or a decimal such as 3.5. The game needs at least 3 whole doors, so the tool returns nothing for those values.
  • Reading 100% as the sum of switch and stay. They do not add up to 100% for N above 3 because the other doors still hold probability that you could not choose.

About this calculator

The Monty Hall problem, named after the host of the game show Let's Make a Deal, is a famous probability puzzle: a contestant picks one of several doors, the host — who knows what's behind each door — opens a losing door, and the contestant is offered the chance to switch. Counterintuitively, switching is almost always the better strategy: with three doors, staying wins with probability 1/3 while switching wins with probability 2/3. This calculator generalizes the puzzle to any number of doors, showing exactly how the switching advantage shrinks (but never disappears) as more doors are added.

The problem is a mainstay of probability theory and decision-theory education because it exposes how strongly conditional information — the host's deliberate, non-random choice of which door to open — changes the odds compared to naive intuition. It routinely appears in statistics courses, game theory discussions, and even in classic debates over conditional probability that once split professional mathematicians.

Was this helpful?

You can pick up to 3 reactions.

Comments

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators