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Subject: The Computer-Chess Player And The Mathematician (was: Waltzing Matilda

Author: fca

Date: 04:26:14 08/18/98

Go up one level in this thread


On August 17, 1998 at 10:17:11, Dan Homan wrote:

>It seemed clear to me that you defined the trials as separate units, on
>re-reading I see that "20 in a row" could stretch across trials.

As Dann Corbitt pointed out :-)

>If you would like to work out this case, let me know the answer.

Leaving aside the end-effects (re first 19 and last 19 trials), it is still a
very unpleasant problem.

>You asked me to reconsider my answer for (c), but I didn't answer (c).
>In c) it is quite ambiguous what you meant by "the chance this result does
>not contradict the fair coin hypothesis", so I simply gave you the
>probability of getting 4 trials of "20 in a row" in an N-trial experiment.

You wrote:

(N!/(4!(N-4)!)) x (2 x (1/2)^10)^4 x (1 - 2 x (1/2)^20)^(N-4)

while I believe you meant:

(N!/(4!(N-4)!)) x (2 x (1/2)^20)^4 x (1 - 2 x (1/2)^20)^(N-4)

:-)

>My "off-my-head" statistics knowledge is not good enough to answer such
>an ill-posed question

This is an absolutely correct observation by Don, IMO.  Let me expand:

The question was (deliberately) a "bad question" - and that was the main point
of the post.  Statisticians will be familiar with "bad questions".

By a *bad question* is _not_ meant an ambiguous question.  I sort-of define it
later.

I already gave an example of a *bad question* (I repeat, with some
simplification):

Given: A child may be only a boy (B) or a girl (G). Twins and higher do not
occur.  Bs and Gs are equiprobable.  The sex of any existing child does in no
way affect the probability of the sex of its later-born sibling or half-sibling
being B or G.
A man has exactly two biological children.
Question (1): "One of them is a B. What is the probability that both are Bs?"
Question (2): "The older of them is a B. What is the probability that both are
Bs?"

Now the answer to (1) is 1/3, and to (2) is 1/2.

(1) is a *bad* question.  The "correct" answer to a bad question cannot ever
help.
1/2 is obviously the correct answer TO THE QUESTION THAT *SHOULD* BE ASKED.

This is non-trivial and non-semantic.

A better question (c) could be (?) be "What is the probability that the coin is
fair, given just this result?"

Please try that, as it has direct ches bearing.

>In (d) I simply set the dispersion of a N-trial result equal to 1 and solved
>for N.  This is a ball-park figure, I know, but it was the best I could
>do sitting at the computer screen.  If you have a better result, please
>let me know.

Again, the question's quality should be examined.

Bruce's answer has a lot to recommend it.

>This was obviously more of an exercise in word games than statistics.

Not really.  It was to illustrate that there is such a beast as a "bad question"
in statistics.
I spotted several bad questions in the parent thread here....

>I'm not sure what point you are making

As stated :-)

>except

"in addition to"  ;-)

> the obvious one that these
>questions need to be well defined or we will all talk right past one
>another.

That also - misunderstandings of this sort are endemic.  But the "bad question"
matter is worse.  Hence my post.

>P.S.  Here is an interesting story about physists and mathematicians....

>There was a famous physist (I can't remember his name) who claimed to be
>able to tell a physist from a mathematician by how they answered the
>following question:
>
>  Two brothers live 10 miles apart.  They both leave their houses one day,
>  at the same time, and walk on a straight line towards one another.  Each
>  brother is walking at 4 miles an hour.  One brother has a dog who leaves
>  his house with him.  The dog is running at 8 miles an hour.  When the dog
>  reaches the other brother, he turns around and goes back to his owner.
>  When he reaches his owner, he turns around again towards the other brother.
>  The dog keeps up this back and forth travel between the brothers until they
>  meet.  How far has the dog traveled?
>The fellow found that physists will reply with the correct answer instantly,

viz. 8 = Dog Speed = Man speed relative to other man = 4 + 4, hence dog travels
as much as men did i.e. 10 miles.

>while mathematicians will take several minutes to sum the series before
>giving the correct answer.  He found this to be an excellent descriminator
>between physists and mathematicians.  One day, this famous physist met the
>reknown mathematician, Jon Von Neumann, to whom he posed this question.  Von
>Neuman replied instantly, which surprised the physist.  He asked Von Neumann,
>"But you are a mathematician, I was sure you would sum the series."
>To which, Von Neumann replied, "I did."

Well-known story, not quite confirmed in a recent biography of JvN.

>A physist, mathematician, and engineer are given the challenge of enclosing
>the most area with the shortest length of fence.  The engineer promptly
>forms a circle, stating that it is the most efficient solution.  The physist
>takes the fence and starts making a straight line toward the horizion
>while mumbling something about one infinity being larger than another.
>The mathematician thinks for a moment and then makes a very tight circle of
>fence around himself and states, "I define myself to be in the outside!".

lol!  Never heard that one before... The chess player was too busy hopping
around mimicking knight moves, presumably... ;-)

Here are some others... you will not have heard them before in entirety, as I
have added several persons to each.

An engineer, a chemist, a mathematician, a philosopher and a computer-chess
player are marooned on a desert island
with only a tin of pilchards and no can opener. The engineer tries to
make a tin opener out of driftwood and pebble chips, but fails to
break into the tin. The chemist tries to make hydrochloric acid from
seawater, but the acid is too weak to eat through the metal. The
mathematician picks up the tin and stares intently at and then says
"let us assume that the tin is open".  The philosopher smiles and says "Those
were nice pilchards".
The computer chess player wakes up from his reverie at the philosopher's words
and says "May I have the dregs in the can, and the HCl, so I can make a battery
to power my computer as I need to examine the Latvian".

or

Five men share a train compartment.  An Astronomer, a Physicist, a statistician,
a computer-chess-player, a Mathematician and a philosopher in one little room.
Glancing out the window, the Astronomer noticed a flock of sheep grazing and
remarked "ALL sheep are white".  The Physicist quietly viewed the scene and
remarked "SOME of the sheep are white".  Punted the statistician "There is at
least one white sheep and I shall apply Bayes Theorem to deduce the likelihood
that other sheep are white." The mathematician briefly glanced up from his
copy of AMS and asserted that "There is at least one sheep that is white on at
least one side". The philosopher came in with "I can't prove that sensations
have sources, but if some of them do, then I may have a mind and a body; thus I
may exist. Further, if some of the other sensations I am receiving have sources,
I may be in a railroad car. There may be one or more other people in the car.
There may be a window in the car. And outside of such window there may be a
place where one or more things which might be called sheep may exist that are
white if white exists on at least one side of their bodies, if they have bodies.
I need to make a few more assumptions, but it'll all fit on a sheet of A4".
The computer chess player looked up from his Travel Novag, disturbed by all the
conversation, and said "No... a4 fails-low at 6-ply full-width"

;-)

In computer-chess

Kind regards

fca

PS: Yes I can count.  The mathematician was a woman.

;-)



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