[FRIAM] [EXT] the Monty Hall problem

Fri Aug 11 12:01:13 EDT 2023

In *Stella Maris*, Cormac Macarthy's new novel depicting  the inner life of
a female polymaths, the author has this wonderful quote concerning the
relation between mathematical intuition and mathematical proof:

*The core question is not how you do math but how does the unconscious do
it.  How is it that it's demonstrably better at it than you are? You work
on a problem and then you put it away for a while.  But it doesnt go away.
It reappears at lunch.  Or while you're taking a shower.  It says: Take a
look at this.  What do you think? Then you wonder why the shower is cold.
Or the soup.  Is this doing math? I'm afraid it is. How is it doing it? We
dont know.  I've posed the question to some pretty good mathematicians.
How does the unconscious do math?? Some who'd thought about and some who
hadnt. For the most part they seemed to think it unlikely that the
unconscious went about it in the same way we did.   ....A few thought that
if it had a better way of doing mathematics it ought to tell us about it. *
*We**ll, m**aybe.  **Or maybe it thinks were not smart enough to understand
it  ...*

*Sometimes you get a clear sense that* *doing math is largely just feeding
data into*
* the substation and waiting to see what comes out. *

I suspect that  *Stella Maris*, with it's brother-sister relation, owes
something to The *Weil Conjecture*, a book about the tortured philosopher
Simone Veil and her mathematician brother Andrew.  (I owe my reading of
both of these books to Jon Zingale who is, right now, probably, writhing in
pain at the chipper insouciance with which I am talking about them.  I
apologize, Jon)  My brother was a mathematician, and I can remember him
churning out scrolls of yellow lined paper covered with mysterious inked
symbols while the babies wailed around him.  I suspect that for every
mathematician, there is a flock of relatives, trying to reach across the
chasm.

Speaking of children, mine are leaving on tuesday.  Perhaps we can get some
chess in after that.

NIck

Nick

On Fri, Aug 11, 2023 at 9:46 AM John Kennison <JKennison at clarku.edu> wrote:

> Hi Nick,
>
> I think you are onto something with the "intuition trap". When I first
> heard the Monty Hall problem, I suspected the best strategy would be to
> stick to one's original choice. If Monty Hall is trying to get me to change
> my choice, he is probably trying to avoid having to give me an expensive
> car.
>
> A mathematical proof requires nothing but cold logic. Finding a proof
> usually requires intuition.
>
> --John
>
> ------------------------------
> *From:* Friam <friam-bounces at redfish.com> on behalf of Nicholas Thompson <
> thompnickson2 at gmail.com>
> *Sent:* Wednesday, August 9, 2023 10:46 PM
> *To:* The Friday Morning Applied Complexity Coffee Group <
> friam at redfish.com>
> *Subject:* [EXT] [FRIAM] the Monty Hall problem
>
> In a  moment of supreme indolence [and no small amount of arrogance] I
> took on the rhetorical challenge of explaining the correct solution of the
> Monty Hall problem (switch).   I worked at it for several days and now I
> think it is perfect.
>
> *The Best Explanation of the Solution of the Monty Hall Problem*
>
> Here is the standard version of the Monty Hall Problem, as laid out in
> Wikipedia:
>
> *Suppose you're on a game show, and you're given the choice of three
> doors: Behind one door is a car; behind the others, goats. You pick a door,
> say No. 1, and the host, who knows what's behind the doors, opens another
> door, say No. 3, which has a goat. He then says to you, "Do you want to
> pick door No. 2?" Is it to your advantage to switch your choice?*
>
> This standard presentation of the problem contains some sly “intuition
> traps”,[1] <#m_-8597096738618129438_m_9217272474254335166_x__ftn1> so put
> aside goats and cars for a moment. Let’s talk about thimbles and peas.  I
> ask you to close your eyes, and then I put before you three thimbles, one
> of which hides a pea.  If you choose the one hiding a pea, you get all
> the gold in China.  Call the three thimbles, 1, 2, and 3.
>
> 1.        I ask you to choose one of the thimbles.  You choose 1.  What
> is the probability that you choose the pea.   ANS: 1/3.
>
> 2.       Now, I group the thimbles as follows.  I slide thimble 2 a bit
> closer to thimble 3 (in a matter that would not dislodge a pea) and I
> declare that thimble 1 forms one group, A, and thimble 2 and 3 another
> group, B.
>
> 3.       I ask you to choose whether to *choose from* Group A or Group B:
> i.e, I am asking you to make your choice of thimble in two stages, first
> deciding on a group, and then deciding which member of the group to pick.
> Which *group* should you choose from?  ANS: It doesn’t matter.   If the
> pea is in Group A and you choose from it, you have only one option to
> choose, so the probability is 1 x 1/3.  If the pea is in Group B and you
> choose from it, the pea has 2/3 chance of being in the group, but you must
> choose only one of the two members of the group, so your chance is again,
> 1/3:  2/3 x ½ = 1/3.
>
> 4.       Now, I offer to guarantee you that, if the pea is in group B,
> and you choose from group B, you will choose the thimble with the pea.
> (Perhaps I promise to slide the pea under whichever Group B thimble you
> choose, if you pick from Group B.)  Should you choose from Group A or
> Group B?   ANS:   Group B.  If you chose from Group A, and the pea is
> there, only one choice is possible, so the probability is still 1 x 1/3=1/3.
> Now, however, if you chose from group B, and the pea is there, since you
> are guaranteed to make the right choice, the probability of getting the pea
> is 1 x 2/3=2/3.
>
> 5.       The effect of Monty Hall’s statement of the problem is to sort
> the doors into two groups, the Selected Group containing one door and the
> Unselected Group, containing two doors.   When he then shows you which
> door in the unselected group does not contain the car, your choice now
> boils down to choosing between Group A and Group B, which, as we have known
> all along, is a choice between a 1/3 and a 2/3 chance of choosing the group
> that contains the pea.
>
> ------------------------------
>
> [1] <#m_-8597096738618129438_m_9217272474254335166_x__ftnref1> The
> intuition trap has something to do with the fact that doors, goats, and
> cars are difficult to group.  So, it’s harder to see that by asking you
> to select one door at the beginning of the procedure, Monty has gotten you
> the group the doors and take the problem in two steps.  This doesn’t
> change the outcome, but it does require us to keep the conditional
> probabilities firmly in mind. “IF the car is in the unselected group, AND I
> choose from the unselected group, and I have been guaranteed to get the car
> if I choose from the unselected group, THEN, choosing from the unselected
> group is the better option.”
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