What's a boy to do?
 

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.

About the only thing I can figure is that it is much like many threads
on ROFF in that most folks, myself included at times, don't always
_read_ what they are "reading," but rather, um, infer from what is
written by what they _think_ is being said.

No doubt many ROFFians are guilty of that; but that is also the intentional
nature of "brain teasers".

In this case, they are
simply ignoring that there are 3, not 2, boards and therefore, the
chances cannot be 1 in 2.

Not exactly, but similar IMO. There are, in fact, 2 choices remaining. The
failure is that of seeing the two choices as random for the original player.
Most folks, I expect, see the problem from the perspective of your "third
person" for whom they *are* random.

Joe F.

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.

What's a boy to do?
 

"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

What's a boy to do?
 

"Opus McDopus" wrote in message
...

"asadi" wrote in message
...

If I'm one of the remaining members I'd say your chances were pretty
damned good....

john

Are we goin to remain on non-speaking terms forever?

Op


okay, I'll give up in a couple of hours and try sending and e-mail and we'll
figure out why yours are bouncing...


I'd do it now but I'm busy making sure Vince wins the tontine...

john

What's a boy to do?
 

"asadi" wrote in message
. ..

okay, I'll give up in a couple of hours and try sending and e-mail and
we'll figure out why yours are bouncing...


I'd do it now but I'm busy making sure Vince wins the tontine...

john

You can get me at: or you can call me at:
828-292-9005

I'm always on speakin' terms with you, as far as you know, anyway :~^ )

Don't know why your e-mails would have bounced, but I will check my
kill-file?

Op

What's a boy to do?
 

"Opus McDopus" wrote in message
...

Don't know why your e-mails would have bounced, but I will check my
kill-file?

Op

Nope, you aren't in Kill-file, just the usual suspects?

Op

What's a boy to do?
 

"Jonathan Cook" wrote in message
...
riverman wrote:

You toss three darts at a target. Dart A misses the target, then Dart B
misses by even more. What is the probability that Dart C will miss by
more
than Dart A?

Not enough information. But _don't_ play darts barefoot if you
are prone to dropping things...

Jon.

Wrong on the first count, right on the second. :-)

--riverman

What's a boy to do?
 

Jonathan Cook wrote:
riverman wrote:

You toss three darts at a target. Dart A misses the target, then Dart B
misses by even more. What is the probability that Dart C will miss by more
than Dart A?

Not enough information.

That sounds right to me. However, under the circumstances I can't
shake a nagging suspicion that there's a bad smell somewhere in there
that I haven't noticed yet.

To put it another way, while a given mathematical proposition may be
clear, correct, simple and unambiguous, it does not necessarily follow
that it will be easy (or even possible.....perhaps) to deal with it
clearly, correctly, simply and unambiguously in plain English.....or
any other natural language, for that matter. One of the beauties of
the Monty Hall problem is that it illustrates this divide very nicely.
Each individual who has provided a "real world" example of how to
arrive at the correct solution and why it IS correct undoubtedly feels
that it clear and easy to follow. The fact that several of us have
taken on the task suggests otherwise.

Following, as it does, closely on the heels of the MH problem, this one
is naturally suspect. :)

All of this points to a very fertile field of enquiry that is
characterized (as well as by anything else, I think) by the fact that a
lot of very bright people find anything in mathematics beyond
elementary arithmetic and perhaps a bit of algebra to be a completely
impenetrable mystery. Much of it IS (and has been at least since the
invention of zero) counterintuitive. Anyone who doubts this would do
well to consult Whitehead and Russell.

But _don't_ play darts barefoot if you
are prone to dropping things...

Don't play darts barefoot......period.

Wolfgang

What's a boy to do?
 

Wolfgang wrote:
Jonathan Cook wrote:
riverman wrote:

You toss three darts at a target. Dart A misses the target, then Dart B
misses by even more. What is the probability that Dart C will miss by more
than Dart A?

Not enough information.

That sounds right to me. However, under the circumstances I can't
shake a nagging suspicion that there's a bad smell somewhere in there
that I haven't noticed yet.


A nice paraphrase of what I tell my students: some things are simple if
you know how, and impossible if you don't.

To put it another way, while a given mathematical proposition may be
clear, correct, simple and unambiguous, it does not necessarily follow
that it will be easy (or even possible.....perhaps) to deal with it
clearly, correctly, simply and unambiguously in plain English.....or
any other natural language, for that matter. One of the beauties of
the Monty Hall problem is that it illustrates this divide very nicely.
Each individual who has provided a "real world" example of how to
arrive at the correct solution and why it IS correct undoubtedly feels
that it clear and easy to follow. The fact that several of us have
taken on the task suggests otherwise.

One reason that math terminology exists (other than to exclude, as is
with all jargon), is because many things can be explained unambiguously
in math to others who speak the language, and hopefully solved in that
terminology. If it were possible to put everything into non-math terms
easily, we would need neither math terminology nor prerequisite
knowledge.


Following, as it does, closely on the heels of the MH problem, this one
is naturally suspect. :)

All of this points to a very fertile field of enquiry that is
characterized (as well as by anything else, I think) by the fact that a
lot of very bright people find anything in mathematics beyond
elementary arithmetic and perhaps a bit of algebra to be a completely
impenetrable mystery. Much of it IS (and has been at least since the
invention of zero) counterintuitive. Anyone who doubts this would do
well to consult Whitehead and Russell.

No argument that math can be challenging, although I'm not sure what
you mean by 'bright people'. It takes training to learn math...that old
'Royal Road' thing, but people want it to be intuitive and I suspect
that this is and has always been the core of the problem.

But the thing the MH puzzle does, as well as this one, is it makes
people who are not well-versed in math but who feel like their
intuition is the sacred measure, face their error. When faced with the
contradiction between their instincts and the mathematical reality of
the MH puzzle, only a fool would insist that the math is wrong and
their instincts are correct. Yet many very bright people are fools. :-)

But, as with many things mathematical (especially many things
statistical), the key to the answer is in how you approach the
solution. To answer the darts question, merely rely on the definition
of probablilty: the number of ways to achieve your objective, divided
by the number of possible outcomes. List all the possible arrangements
of how the darts could land, and count how many fit our scenario.
First, list all the ways to throw three darts, A B and C.

ABC
ACB
BAC
BCA
CAB
CBA

Remember, we are looking at a conditional probability; dart B has
already landed farther than dart A. So our list of outcomes is limited
to:

ABC
ACB
CAB

Our 'definition of success' is when dart C lands further than dart A,
which is clearly only the first two arrangements. So the probability of
throwing a third dart that lands farther than the first (given the
second dart has already landed farther than the first), is 2/3.

Its an unsettling conclusion, because people want to make the argument
that the distance from the bullseye affects the probability of each
outcome. However, every possible distance affects every outcome
equally, so they are all still equally likely, as counterintuitive as
it may be.

--riverman

What's a boy to do?
 

riverman wrote:
Wolfgang wrote:
Jonathan Cook wrote:
riverman wrote:

You toss three darts at a target. Dart A misses the target, then Dart B
misses by even more. What is the probability that Dart C will miss by more
than Dart A?

Not enough information.

That sounds right to me. However, under the circumstances I can't
shake a nagging suspicion that there's a bad smell somewhere in there
that I haven't noticed yet.


A nice paraphrase of what I tell my students: some things are simple if
you know how, and impossible if you don't.

Not to put too fine a point on it, but that would not be some
things......that would be pretty much everything.

To put it another way, while a given mathematical proposition may be
clear, correct, simple and unambiguous, it does not necessarily follow
that it will be easy (or even possible.....perhaps) to deal with it
clearly, correctly, simply and unambiguously in plain English.....or
any other natural language, for that matter. One of the beauties of
the Monty Hall problem is that it illustrates this divide very nicely.
Each individual who has provided a "real world" example of how to
arrive at the correct solution and why it IS correct undoubtedly feels
that it clear and easy to follow. The fact that several of us have
taken on the task suggests otherwise.

One reason that math terminology exists (other than to exclude, as is
with all jargon), is because many things can be explained unambiguously
in math to others who speak the language, and hopefully solved in that
terminology.

Sure.

If it were possible to put everything into non-math terms
easily, we would need neither math terminology nor prerequisite
knowledge.

True, we wouldn't need a specialized terminology. Prerequisite
knowledge strikes me as a bit ambiguous......could mean any of several
things. Any way I look at it, though, I can't see a way to do without
it.

Following, as it does, closely on the heels of the MH problem, this one
is naturally suspect. :)

All of this points to a very fertile field of enquiry that is
characterized (as well as by anything else, I think) by the fact that a
lot of very bright people find anything in mathematics beyond
elementary arithmetic and perhaps a bit of algebra to be a completely
impenetrable mystery. Much of it IS (and has been at least since the
invention of zero) counterintuitive. Anyone who doubts this would do
well to consult Whitehead and Russell.

No argument that math can be challenging, although I'm not sure what
you mean by 'bright people'.

Well, take several PhDs vehemently defending the wrong answer to a
particular simple problem, for example. As a class, PhDs are arguably
reasonably bright people. More generally, I guess I mean people who
show an aptitude for dealing with a variety of different kinds of
problems and an ability to articulate their thoughts in a manner that
is comprehensible to other concerned parties.

It takes training to learn math...that old
'Royal Road' thing, but people want it to be intuitive and I suspect
that this is and has always been the core of the problem.

To the extent that people are more interested in solutions than
problems (and I think it's fair to say that most people probably are in
most instances), I suspect that they more often than not would prefer
that the answers come intuitively rather than at the expense of hard
mental labor. This seems to me like a perfectly reasonable position.
Personally, I don't think this is "the" core of the problem that so
many people have with math. I believe it's more complicated than that.
For one thing, people start out as children, naturally and voraciously
curious little beasts.......intellectual sponges eager to suck up
whatever they can. Despite some minor variations among different
tribes and cults, educational institutions and, more particularly, the
cultures they serve tend to be remarkably indistinguishable in their
capacity to crush that curiosity at a tender age......as they have
done, for the most part, for centuries. There are many other problems.

But the thing the MH puzzle does, as well as this one, is it makes
people who are not well-versed in math but who feel like their
intuition is the sacred measure, face their error.

In theory. In practice, most just brush it aside. Ever been to Las
Vegas?

When faced with the
contradiction between their instincts and the mathematical reality of
the MH puzzle, only a fool would insist that the math is wrong and
their instincts are correct.

The Monty Hall problem is simple enough that trusting to intuition is
easily demonstrable as a foolish course of action. That's what makes
it illustrative. That's what makes it interesting. But this is by no
means always the case. Take a look at competing sophisticated economic
theories on the advisability of debt load sometime. Whose math are you
going to believe? Intuition tells me that having more cash than debt
is a good thing.

Yet many very bright people are fools. :-)

All bright people do foolish things, but they are NOT fools.....by
definition.

But, as with many things mathematical (especially many things
statistical), the key to the answer is in how you approach the
solution. To answer the darts question, merely rely on the definition
of probablilty: the number of ways to achieve your objective, divided
by the number of possible outcomes. List all the possible arrangements
of how the darts could land, and count how many fit our scenario.
First, list all the ways to throw three darts, A B and C.

ABC
ACB
BAC
BCA
CAB
CBA

Remember, we are looking at a conditional probability; dart B has
already landed farther than dart A. So our list of outcomes is limited
to:

ABC
ACB
CAB

Our 'definition of success' is when dart C lands further than dart A,
which is clearly only the first two arrangements. So the probability of
throwing a third dart that lands farther than the first (given the
second dart has already landed farther than the first), is 2/3.

Its an unsettling conclusion, because people want to make the argument
that the distance from the bullseye affects the probability of each
outcome. However, every possible distance affects every outcome
equally, so they are all still equally likely, as counterintuitive as
it may be.

I don't see anything in the above about whether the person throwing the
darts is a champion player.....or drunk.....or blind. Remember, we are
looking at a conditional probability; we know very little about the
conditions that apply. At its root, this isn't really a math problem.

Counterintuitive, nicht wahr?

Wolfgang
to the man with a hammer.......