probability
did you know switching doors can double your chances?
one prize, three doors, and a host who knows more than you do.
⅓ → ⅔
stay → switch
switching wins whenever your first choice was wrong, under the standard host rules.
- pick one door
- host reveals a loser
- switch doors
the idea
pick one of three doors. one hides a prize. a host who knows the answer opens a different, losing door and always offers a switch. switching wins two-thirds of the time; staying wins one-third.
how it works
if you picked the prize first, switching loses. if you picked either of the two losing doors, the host must expose the other loser, so switching wins. two of the three equally likely starting situations favour switching.
go deeper
what did your first choice actually buy?
pick one of three doors. one hides a prize. a host who knows the answer opens a different, losing door and always offers a switch. switching wins two-thirds of the time; staying wins one-third.
your first pick had a one-in-three chance of being right. the two doors you left behind collectively had two-thirds of the chance. the host cannot simply erase that advantage by revealing a known loser.
follow the wrong first choices
if you picked the prize first, switching loses. if you picked either of the two losing doors, the host must expose the other loser, so switching wins. two of the three equally likely starting situations favour switching.
imagine 100 doors instead. you choose one, then the informed host opens 98 losing doors. the one remaining alternative suddenly looks much more interesting.
why is the answer not fifty-fifty?
the host's action depends on where the prize is. it is not a random deletion of a door. your original choice stays right in one of the three starting cases.
P(win by switching) = P(first pick was wrong) = 2/3
which rules make this work?
- the host must know the prize location, always reveal a loser, and always offer the switch.
- a host who sometimes opens the prize or selectively offers a switch creates a different probability problem.
sources & further reading
concepts: conditional probability · Monty Hall · information
2 minute read