The Monty Hall Problem & Conditional Probability
Steve Selvin / Marilyn vos Savant · 1975
"Switching doors in Monty Hall doubles your chance of winning from 1/3 to 2/3 — because the host's action of opening a losing door is not random, it leaks information about where the prize isn't."
You pick one of three doors. The host, who knows what's behind each, opens a different losing door and offers you the switch. Intuition screams 50/50. The math says switch and you win two-thirds of the time.
Initially you have a 1/3 chance of having chosen the winner and 2/3 that the prize is behind the other two. The host's non-random reveal collapses those two doors into one — but the probability mass stays with the unchosen set. Conditioning on the host's forced move, switching implements the 2/3 branch. Generalizes: ignore how information was generated and you mis-update; 'new evidence' is only as informative as the process that produced it.
Why does switching win 2/3 of the time in the Monty Hall problem?
Read more about the topic
The explanation above is written with AI assistance. These are the originals — go to them to check it.
- Monty Hall problemWikipedia
P vs NP & Computational Complexity
"Cook and Levin formalized the most important open question in computer science: if a solution can be verified quickly (NP), can it also be found quickly (P)? Most experts believe no — and that gap explains why many optimization, routing, and scheduling problems have no efficient perfect solution."