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The Monty Hall Problem
You're on a TV game show. In front of you are three doors:
there's a great prize behind one door, and nothing behind the other two. You choose a door.
Then the host (Monty Hall) opens one of the two doors you didn't choose to show
that there is nothing behind that door. It would be bad for the TV ratings if he
opened the prize door: you'd know you had lost and the game would be over; so Monty knows
where the prize is, and he always opens a door that doesn't have a prize behind it (Monty is
Canadian, so you know you can trust him). You're now facing two unopened doors,
the one you originally picked and the other one, and the host gives you a chance to change
your mind: do you want to stick with the door you originally chose, or do you want to switch
to what's behind the other door?
The Question
What should you do to have the best chance of winning?
The Answers
There are only three answers:
a) stick with the door you originally chose,
b) switch to the other door, or
c) it makes no difference what you do.
There are lots of different arguments for each of those three answers.
Twenty-Five Arguments
Here are twenty-five arguments for and against different answers.
Some of the arguments are not in favour of any particular answer,
but are directed against supposedly bad arguments for one of the answers.
The first five arguments are from a series of letters written to Marilyn vos Savant,
who holds the world's highest I.Q. and who first publicized the problem in her
magazine column Ask Marilyn. Her answer leads off.
1. SWITCH (Marilyn's answer)
...you should switch. The first door has a 1/3 chance of winning,
but the second door has a 2/3 chance. Here's a good way to visualize
what happened: Suppose there are a million doors, and you
pick door number 1. Then the host, who knows what's behind the
doors and will always avoid the one with the prize, opens them
all except door number 777,777. You'd switch to that door pretty
fast, wouldn't you?
Marilyn vos Savant
2. NO DIFFERENCE (a mathematician)
Dear Marilyn:
Since you seem to enjoy coming straight to the point, I'll do
the same.... you blew it! Let me explain. If one door is shown
to be a loser, that information changes the probability of either
remaining choice, neither of which has any reason to be more
likely, to 1/2. As a professional mathematician, I'm very
concerned with the general public's lack of mathematical skills.
Please help by confessing your error and in the future being more
careful.
Robert Sachs, Ph.D.
George Mason University
3. NOT NO DIFFERENCE (Marilyn's reply)
My original answer is correct. But first, let me explain why your
answer is wrong. The winning chances of 1/3 on the first choice
can't go up to 1/2 just because the host opens a losing door.
To illustrate this, let's say we play a shell game. You look away,
and I put a pea under one of the three shells. Then I ask you
to put your finger on a shell. The chances that your choice contains
a pea are 1/3, agreed? Then I simply lift up an empty shell from
the remaining other two. As I can (and will) do this regardless
of what you've chosen, we've learned nothing to allow us to revise
the chances on the shell under your finger.
Marilyn vos Savant
4. NOT NOT NO DIFFERENCE (reply to the reply)
Dear Ms. vos Savant:
It is apparent from your "Ask Marilyn" column... that
being smart is no guarantee of being correct. Your analysis of
the game-show probabilities... reveals a misunderstanding of the
rudiments of probability theory, and an appalling lack of logic...
Consider first the shell game, in which a single pea is placed
under one of three shells. Let's label the shells A, B, and C
and suppose that one player, named Abe, puts his finger on shell
A and a second player, named Ben, puts his finger on shell B.
In the absence of any information of the location of the pea (other
than the facts that there is only one pea and it is under one
of the three shells), we assign a value of 1/3 to the probability
of the pea being under any one shell: pA = pB = pC = 1/3. (Note
that the probabilities must sum to 1.) Now suppose that shell
C is lifted and the pea is not seen. We now know that pC = 0.
According to your argument, now pA = 1/3 and pB = 2/3 because
"nothing has been learned to allow us to revise the odds"
on the shell under Abe's finger. But by the same reasoning, pB
should remain 1/3! Now either 1/3 + 1/3 = 1 or your argument is
wrong. I leave it to you to figure out which.
...I urge you to lower your mantle of omniscience and (following
the lead of Ann Landers) seek the advice of experts when the subject
matter is outside your area of expertise. Your ignorant responses
are hurting the fight against mathematical illiteracy.
Sincerely,
David Loper
Professor of Mathematics
Florida State University.
5. NO DIFFERENCE (The Engineer's analogy)
Imagine three runners in the 100 meter dash. The runners are
so evenly matched that the odds of any given runner winning are
completely random. I am asked to guess the winner and select runner
number 1. After making my choice, runner number 3 pulls a hamstring
and cannot compete. Based on your [Marilyn's] answer to the game
show problem, I would conclude that the unfortunate injury to
runner number 3 has somehow inexplicably made runner number 2
the clear favorite over my original choice. By way of the above
example, I hope it is now clear that your analysis of the game
show problem is flawed.
Jeffrey Hoyt
Department of Mechanical Engineering
Washington State University.
6. NOT SWITCH (Can't trust Female Logic)
Dear Marilyn:
I still think you're wrong.
There is such a thing as female logic.
Don Edwards,
Sunriver, Oregon
7. SWITCH (Female Logic)
There is such a thing as female logic. Since Marilyn is female, and thinks it's logical to switch, then that's the right answer.
8. SWITCH (the odds changed)
When you first chose a door you had a 1/3 chance of picking the
winner. Now that there are only two doors, the other door has
a 1/2 chance of winning. So switching raises your chance of winning
from 1/3 to 1/2. You should switch.
9. NO DIFFERENCE (the odds changed)
When you first chose a door you had a 1/3 chance of picking the
winner. Now that there are only two doors, the other door has
a 1/2 chance of winning. But that means the door you first picked
also has a 1/2 chance of winning. So there's no point sticking
or switching, since your chance of winning is 1/2 with either
door.
10. SWITCH (two-for-one)
After the host opens a door and you have to decide whether to
stick or switch, your choice is the same as if you could stick
with your door or take both of the other two doors (one
of which, admittedly, has nothing behind it). So you ought to
switch, since your chance of winning is better if you get to take
two doors rather than just one. In fact, you're twice as likely
to win if you switch.
11. STICK (two-for-one)
After the host opens a door and you have to decide whether to
stick or switch, your choice is the same as if you could switch
to the unopened door, or stick with your door and take
what's behind the opened door (which, admittedly, has nothing
behind it). So you ought to stick, since your chance of winning
is better if you get to take two doors rather than just one. In
fact, you're twice as likely to win if you stick.
12. SWITCH (The Experiment)
I tried this in an earlier workshop using Dixie cups: sticking
won 93 times, switching won 132 times. According to this, if
you switch you have a 132/225 chance of winning (about 59%),
versus a 93/225 chance of winning if you stick (approx. 41%);
so you're 18% more likely to win if you switch.
So you should switch.
13. NOT SWITCH (not-a-reliable experiment)
While walking around watching people play the game/experiment, I noticed
several of the groups playing the game not-quite the way it was supposed to
be played (e.g. lifting the first Dixie cup the contestant pointed to, forgetting what
cup the prize was hidden under). That means we can't really trust the results
cited above; so this experiment was too unreliable to show that you should
switch.
14. NO DIFFERENCE (It's a guess)
We're assuming the game isn't rigged, and the contestant isn't
psychic or can't see through solid objects, right? So the contestant
has no information which would help her know where the prize is.
So the contestant has to guess. There are two doors, so
either guess will be 50:50. So, it makes no difference which door
the contestant chooses.
15. STICK (It's a guess, so go with your first instincts)
We're assuming the game isn't rigged, and the contestant isn't
psychic or can't see through solid objects, right? So the contestant
has no information which would help her know where the prize is.
So the contestant has to guess. There are two doors, so
either guess will be 50:50. So you can't improve your chance
of winning by sticking or switching, so you should stick with
your first choice because it's important to trust your gut instinct.
16. SWITCH (switch from a likely loser)
When you first choose a door you have a 1/3 chance of choosing
the prize door, and a 2/3 chance of picking a loser. If you've picked the prize door,
then sticking wins. If you pick a loser, switching wins. Since you're twice as likely
to have picked a loser, you're twice as likely to win if you switch.
17. NO DIFFERENCE (Analogy to Newton's Law)
Newton proved that objects at rest remain at rest unless an outside force acts upon
them, which is like what happens on stage during the game. Once the stage is setup,
with the prizes behind the doors, nobody moves anything around, so everything stays
the same as when they started. At the start each door is equally likely to be the
winner (1/3). The host opens one door, but doesn't do anything to either of the
others. If you treat things that are the same the same way, then they will still be
the same. So the remaining two doors must still be equally likely to be the winner;
so it's 50:50 and sticking or switching doesn't improve your odds of winning.
18. NO DIFFERENCE (Zen answer)
It makes no difference which you choose. If you desire to win,
you have already lost.
The Demo
You can play a computerized version of the game
NOW!
This demo reqires JavaScript and Java, and does not reliably work with all versions of MS Internet Explorer.
19. SWITCH (The Demo)
I played the demo a bunch of times, and I won more often when I switched.
20. NOT SWITCH (Not a real Demo)
That wasn't a real demo: there were no actual doors with prizes behind them, it was just
a bunch of pictures controlled by some computer program, and you have no idea how that
demo was programmed. Maybe it was programmed to show you the prize behind the
"switch to" door so it would seem like you'd win more often when you switched in a real game.
21. NO DIFFERENCE (Majority Rules)
I've asked this question many times in introductory lectures. Most recently, 119 participants answered. The results were: 38 said you should Stick; 5 said Switch; and 76 thought it made No Difference. That's a solid majority of an intelligent audience, so there is good reason to believe it makes no difference.
22. NOT NO DIFFERENCE (Majority doesn't determine truth)
Just because the majority believes something doesn't mean it's true. Arguing like that is a fallacious appeal to popularity, it's not a good reason to think it makes no difference.
23. NOT `NOT NO DIFFERENCE' (Democracy Rules)
What are you, some kind of communist? Majority rules is good enough for democracy, so it's good enough for this piddly little game. If the majority says "no difference", they're right.
24. NO DIFFERENCE (Majority of arguments rule)
Reasonable people adjust their beliefs to fit the evidence. Judging by this page, there are more arguments for "No Difference" than for the other answers, so there's more reason to believe that it makes no difference. There are only two arguments for Stick. There are seven arguments for Switch, but three counter-arguments against them, making a total of four arguments for Switch. The arguments about No Difference are a little more difficult to count: there are six arguments for No Difference, and only one counter-argument against them, but there is also one counter-counter-argument, so if we split the difference the argument against No Difference should only count for half, which leaves us with 5½ arguments for No Difference.
25. SWITCH (Last Reason is always the Best)
Just as your car keys are always in the last place you look, so it is with reasoning: after a long time thinking about it the last thought is probably the best; the last thought here is that there's a better chance to win if you switch; therefore you should switch.
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