Math · Exploration
The Monty Hall Problem
Why switching doors doubles your chance of winning.
Where the puzzle came from
The puzzle is named for Monty Hall, host of the television game show Let’s Make a Deal. It became widely debated after statistician Steve Selvin wrote about it in 1975, and again after Marilyn vos Savant answered a reader’s version in her 1990 Parade column.
The rules matter
One of three doors holds a prize. You choose one. Monty knows where the prize is, always opens a different losing door, and always offers you the chance to switch. If two losing doors are available, he may choose either one; that detail does not change the result.
Investigate
Why isn’t it 50/50?
Your first door keeps its original chance. The other two doors began as a group. Monty’s informed reveal removes a known loser from that group; it does not make a random choice between the two doors left closed. Switching takes the remaining chance.
All three cases
Assuming you first choose Door 1, staying wins 1 of 3 equally likely cases; switching wins 2 of 3.
A different host, a different problem
If the host opened one of the two unchosen doors at random and happened to reveal a goat, the information would be different. The classic answer depends on Monty’s protocol: he knows the prize location and deliberately reveals a goat.
Simulate
Compare independent classic Monty Hall rounds with each strategy.
Always stayRun a simulation.
Always switchRun a simulation.
100-door intuition
With 100 doors, your original choice has a chance. The other 99 doors hold . After Monty opens 98 known losing doors, would you switch?
Related concepts
Monty’s constrained reveal is a question about conditional information, not probability magically moving between doors.