Please note that since I wrote this article, I have been persuaded that the argument it relates to is wrong (meaning that chrysics and Mathematician and irishsultan and ChalkboardCowboy were all right from the start and I should have listened to them rather than arguing with them). Fortunately, I didn't because, for me at least, this little intellectual journey has been far more interesting than it would have otherwise been.
The correct answer for the scenario as it is worded is not 1/2 but rather 1/3 (meaning that the likelihood of winning as a consequence of staying is 2/3).
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One of my interlocutors recently encouraged me to learn more of conditional probability. While I was pretty sure that I knew a bit about it, this didn’t stop me from trying to expand my knowledge. While I was fossicking around the internet I came across an interesting and related little scenario, one that I have modified below:
The correct answer for the scenario as it is worded is not 1/2 but rather 1/3 (meaning that the likelihood of winning as a consequence of staying is 2/3).
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One of my interlocutors recently encouraged me to learn more of conditional probability. While I was pretty sure that I knew a bit about it, this didn’t stop me from trying to expand my knowledge. While I was fossicking around the internet I came across an interesting and related little scenario, one that I have modified below:
Say that Dusty, Lucky and Ned
are booked to perform for El Monty, who is having a Very Bad Day. At the end of the performance, El Monty calls
over one of his henchmen and says: “I like these guys, these are funny
guys! Just kill two of them!”
The henchman asks which two
and El Monty points out that he doesn’t care: “Just pick two of their names at
random. But don’t tell them which ones! We don’t want them to be sad on their last
night, do we?”
The Three Amigos are then led
away into the dungeon and thrown into a cell.
Because Lucky is played by the
better comedian, they all agree that he should survive. They come up with a plan to swap nametags
once they have wheedled some information.
Lucky then wanders up to the
cell door and strikes up a conversation with the guard who promptly refuses to
tell Lucky whether he is going to get shot in the morning. However, after Lucky performs some great gags
and signs an autograph for the guard’s niece, the guard agrees to provide the
name of one of the other Amigos who is definitely going to be executed in the
morning.
Lucky goes back to the other
guys and tells Ned the sad news.
Should Lucky and Dusty swap
nametags? Why?
Before they get the chance to
decide themselves, however, El Monty is suddenly at the door: “¡Stupido! ¿Did I
not tell you, do not tell them?”
El Monty then takes out a gun
and shoots Ned through the forehead, killing him instantly.
Should Lucky and Dusty swap
nametags? Why?
Now,
how is this related to the Reverse Monty Hall Problem?
Lucky knows that there is a
2/3 likelihood of being executed (equivalent to a pre-existing likelihood of
2/3 of having a goat behind any door)
The guard refuses to tell
Lucky anything about his fate (making Lucky equivalent to the unselected door
and making Dusty and Ned equivalent to the selected doors)
The guard identifies Ned as
one of the two to be executed (equivalent to being told that a goat is behind a specific door)
El Monty kills Ned (making the
probability that Ned will be executed 100%, equivalent to knowing with absolute
certainty that there is a goat behind the door, because it's been opened)
Lucky can swap nametags with
Dusty (equivalent to switching doors)
So, put
it another way, if you are Lucky – do you switch your nametag with Dusty (who
is eager to die so that you may live)?
What is the likelihood of you surviving if you do switch?
If you
think that this is somehow not equivalent to The
Reverse Monty Hall Problem, feel free to explain how.
