What's a boy to do?

rb608 wrote:
wrote in message
I don't see how it's (objectively) counter-intuitive,

There are two remaining choices, switch or don't switch. I think 50:50 is
an easy conclusion to draw from that.

Exactly......and that is precisely what makes the Monty Hall problem
interesting......well, that's a part of it, anyway (more about that in
just a moment). It isn't the math. Hell the math is simple enough
that even I (no math wiz......by ANY stretch of the imagination) have
no trouble at all in understanding and accepting various permutations
of the explanation. Anyone adept at mathematics and who takes a moment
to think it through will invariably come up with the right answer and,
doubtless, find the whole thing rather silly.

Somewhat ironically, it takes a basic knowledge of the fundamental laws
of probability to figure out the wrong answer.....you have to know that
tossing a coin will, in the long run, result in something very close to
half heads, half tails. Anyone who doesn't know this can only
guess.....and is as likely to guess right as wrong.....50:50 chance!
Sweet! There is no doubt in my mind that both Craig Whitaker and
Marylin vos Savant were well aware of this when the former posed the
question and the latter decided to answer it.

At least a couple of people have made references to the rules as I
stated them in posing the problem. In fact, there were NO rules.
There was simply a question about how one should proceed in a precisely
and unambiguously stated situation. Suggestions and speculations about
how to work through more or less similar situations (changing the
"rules") may or may not be interesting in their own right, but they
have nothing whatsoever to do with the original problem. I suspect
that most of them have something or other to do with a certain level of
discomfort engendered by the decidedly counterintuitive correct
solution to the original.

When all is said and done, the whole thing is a trick question. What
makes it exquisitely delicious is that, as stated at the outset, I, the
expositor, was not playing any kind of trick on the player......well
not directly, anyway. No, what tricks the player is his or her own
knowledge of probabilities and a lightning quick recognition of an
absurdly easy problem.

Right, Ken?

O.k., that last bit was just a little unfair. Um......or was it?
After all, Haddon said the pretty much same thing. Did anybody else
see it? A shiny new nickel to the first to point out Haddon's own
sorta nasty little trick. (hint: it's in the quotes......more or less)


Bottom line? The Monty Hall problem really isn't much of a
mathematical puzzle at all. What it IS......in spades......is a
beautifully elegant probe into human psychology!

As for illustrating the logic behind the correct solution, here's my
own humble contribution:

Let us change the scenario a bit. Instead of a single player who gets
to decide whether to change his or her pick after one of the losers is
exposed, let's have TWO players......Toivo and Aino. Toivo gets to
pick one of the three possibilities.....Aino automatically gets the
other two. All three positions are exposed. Any one may be the
winner, but no one should have any difficulty in seeing that the smart
money would bet on Aino. Whether in a single round or in repeated
play, the odds are clearly in his favor to the tune of two to
one.......67% to 33%.....not too roughly. Now, let us suppose that
rather than exposing all the possibilites at once, the expositor turns
over one of the boards at random. How does this change the odds?
Clearly, it has absolutely no effect on the odds. O.k., so, at least
one of Aino's two possibilities HAS TO be a loser.....right? After
all, there are three positions and only one of them is the winner.
Alright, so, if the expositor first turns over one of Aino's
possibilities, which is one of the losers, how does this affect the
odds? Again, it cannot possibly affect the odds......the winner and
both losers are where they are.....NOTHING can affect the fact that
Aino wins two times out of three.....more or less......in the long run.

Now, let's go back to the problem as originally stated. Toivo is the
only player. As long as he sticks with his original choice when given
the option, nothing, in essence, is any different than it was in the
two player game......he loses two times out of three......Aino has
simply become invisible. Obvious......right? Right. O.k., so, what
if Toivo chooses to jump one way one time and another the next? Beats
the **** out of me (and you too, if there's an honest bone in your
body). Ah, but what if Toivo changes his choice EVERY time? Well,
then he quite simply BECOMES Aino!!

Wolfgang
dunkenfeld knew.

"Wolfgang" wrote in message
oups.com...

rb608 wrote:
wrote in message
I don't see how it's (objectively) counter-intuitive,

There are two remaining choices, switch or don't switch. I think 50:50
is
an easy conclusion to draw from that.

Exactly......and that is precisely what makes the Monty Hall problem
interesting......well, that's a part of it, anyway (more about that in
just a moment). It isn't the math. Hell the math is simple enough
that even I (no math wiz......by ANY stretch of the imagination) have
no trouble at all in understanding and accepting various permutations
of the explanation. Anyone adept at mathematics and who takes a moment
to think it through will invariably come up with the right answer and,
doubtless, find the whole thing rather silly.

Somewhat ironically, it takes a basic knowledge of the fundamental laws
of probability to figure out the wrong answer.....you have to know that
tossing a coin will, in the long run, result in something very close to
half heads, half tails. Anyone who doesn't know this can only
guess.....and is as likely to guess right as wrong.....50:50 chance!
Sweet! There is no doubt in my mind that both Craig Whitaker and
Marylin vos Savant were well aware of this when the former posed the
question and the latter decided to answer it.

At least a couple of people have made references to the rules as I
stated them in posing the problem. In fact, there were NO rules.
There was simply a question about how one should proceed in a precisely
and unambiguously stated situation. Suggestions and speculations about
how to work through more or less similar situations (changing the
"rules") may or may not be interesting in their own right, but they
have nothing whatsoever to do with the original problem. I suspect
that most of them have something or other to do with a certain level of
discomfort engendered by the decidedly counterintuitive correct
solution to the original.

When all is said and done, the whole thing is a trick question. What
makes it exquisitely delicious is that, as stated at the outset, I, the
expositor, was not playing any kind of trick on the player......well
not directly, anyway. No, what tricks the player is his or her own
knowledge of probabilities and a lightning quick recognition of an
absurdly easy problem.

Right, Ken?

O.k., that last bit was just a little unfair. Um......or was it?
After all, Haddon said the pretty much same thing. Did anybody else
see it? A shiny new nickel to the first to point out Haddon's own
sorta nasty little trick. (hint: it's in the quotes......more or less)


Bottom line? The Monty Hall problem really isn't much of a
mathematical puzzle at all. What it IS......in spades......is a
beautifully elegant probe into human psychology!

As for illustrating the logic behind the correct solution, here's my
own humble contribution:

Let us change the scenario a bit. Instead of a single player who gets
to decide whether to change his or her pick after one of the losers is
exposed, let's have TWO players......Toivo and Aino. Toivo gets to
pick one of the three possibilities.....Aino automatically gets the
other two. All three positions are exposed. Any one may be the
winner, but no one should have any difficulty in seeing that the smart
money would bet on Aino. Whether in a single round or in repeated
play, the odds are clearly in his favor to the tune of two to
one.......67% to 33%.....not too roughly. Now, let us suppose that
rather than exposing all the possibilites at once, the expositor turns
over one of the boards at random. How does this change the odds?
Clearly, it has absolutely no effect on the odds. O.k., so, at least
one of Aino's two possibilities HAS TO be a loser.....right? After
all, there are three positions and only one of them is the winner.
Alright, so, if the expositor first turns over one of Aino's
possibilities, which is one of the losers, how does this affect the
odds? Again, it cannot possibly affect the odds......the winner and
both losers are where they are.....NOTHING can affect the fact that
Aino wins two times out of three.....more or less......in the long run.

Now, let's go back to the problem as originally stated. Toivo is the
only player. As long as he sticks with his original choice when given
the option, nothing, in essence, is any different than it was in the
two player game......he loses two times out of three......Aino has
simply become invisible. Obvious......right? Right. O.k., so, what
if Toivo chooses to jump one way one time and another the next? Beats
the **** out of me (and you too, if there's an honest bone in your
body). Ah, but what if Toivo changes his choice EVERY time? Well,
then he quite simply BECOMES Aino!!



Interesting illustration of how to visualize the correct strategy, but I
suggest any doubters just make a spinner out of a paperclip and a piece of
paper. Draw a circle, divide it into three 'pizza slices' of approximately
the same area, and hold the clip in the center with a pencil tip. Decide
that one 'pizza slice' is the actual prize, spin the clip to choose your
intitial door, and act out the scenario. The logic behind the '2/3' answer
is instantly and undeniably clear.

Or, to paraphrase Wolfgang's Toivo and Aino situation, the odds of winning
by switching doors after one is revealed is exactly equal to the odds that
your original pick was wrong....2/3.

--riverman