IDEA FOUND // IDEA 145
The Monty Hall Problem
“When the host opens a door using information, the two remaining doors do not automatically become equally likely.”
01 / PLAINLY
What it means, plainly
In the classic problem, you choose one of three doors. The host knows the prize location, always opens another prize-free door, and offers a switch; under those rules, switching gives you a 2/3 chance of winning.
02 / CONTEXT
A little more
If the prize was placed randomly, you choose one door, and the host always opens another prize-free door before offering a switch, the original choice remains at 1/3 and switching wins with probability 2/3.
03 / WHY IT MATTERS
Why it matters
It shows that receiving information does not automatically split probability evenly; the method and reason for the reveal matter.
04 / EXAMPLE
A familiar example
Your first door is right only one time in three; in the other two, the host's rule concentrates the prize behind the only available switch.
05 / LIMIT
What it does not mean
The 2/3 result depends on the host's protocol. If doors are opened blindly or switching is offered only sometimes, the probabilities can differ.
06 / NOTICE
Notice it in your day
Make a table for the three possible prize locations, always choose door 1, and apply the host's rule. Count how often switching wins.
FINAL NOTE
The idea worth keeping
Before splitting probability evenly, model what the person revealing information knew and which rules they followed.
QUESTIONS / 02
Questions people still have
Why does switching win with probability 2/3 in the classic problem?
The initial choice is right only 1/3 of the time; when it is wrong, the host's rule leaves the winning switch available.
Does the result hold for any host behavior?
No. The host must know the prize, always open a losing door, and always offer the switch.