<div dir="ltr"><div>In <i>Stella Maris</i>, 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:</div><div> <br></div><div style="margin-left:40px"><font style="font-family:georgia,serif" size="2"><b><i>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. </i></b></font><font style="font-family:georgia,serif" size="2"><b><i>We</i></b></font><font style="font-family:georgia,serif" size="2"><b><i>ll, m</i></b></font><font style="font-family:georgia,serif" size="2"><b><i>aybe. </i></b></font><font style="font-family:georgia,serif" size="2"><b><i>Or maybe it thinks were not smart enough to understand it ...</i></b></font></div><div style="margin-left:40px"><font style="font-family:georgia,serif" size="2"><b><i><br></i></b></font></div><div style="margin-left:40px"><font style="font-family:georgia,serif" size="2"><b><i>Sometimes you get a clear sense that</i></b></font>
<font style="font-family:georgia,serif" size="2"><b><i>doing math is largely just feeding data into</i></b></font><font style="font-family:georgia,serif" size="2"><b><i> the substation and waiting to see what comes out. <br></i></b></font></div><div style="margin-left:40px"><font style="font-family:georgia,serif" size="2"><b><i><br></i></b></font></div><div><font size="2" face="arial,sans-serif">I suspect that <i>Stella Maris</i>, with it's brother-sister relation, owes something to The <i>Weil Conjecture</i>, 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.<br></font></div><div><font size="2" face="arial,sans-serif"><br></font></div><div><font size="2" face="arial,sans-serif">Speaking of children, mine are leaving on tuesday. Perhaps we can get some chess in after that. <br></font></div><div><font size="2" face="arial,sans-serif"><br></font></div><div><font size="2" face="arial,sans-serif">NIck <br></font></div><div><font size="2" face="arial,sans-serif"><br></font></div><div><font size="2" face="arial,sans-serif">Nick <br></font></div>
</div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Aug 11, 2023 at 9:46 AM John Kennison <<a href="mailto:JKennison@clarku.edu" target="_blank">JKennison@clarku.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div>
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Hi Nick,</div>
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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.</div>
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<span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif;font-size:12pt;color:rgb(0,0,0)">A mathematical proof requires nothing but cold logic. Finding a proof usually requires intuition.</span><br>
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--John</div>
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<div id="m_-8597096738618129438m_9217272474254335166divRplyFwdMsg" dir="ltr"><font style="font-size:11pt;color:rgb(0,0,0)" face="Calibri, sans-serif"><b>From:</b> Friam <<a href="mailto:friam-bounces@redfish.com" target="_blank">friam-bounces@redfish.com</a>> on behalf of Nicholas Thompson <<a href="mailto:thompnickson2@gmail.com" target="_blank">thompnickson2@gmail.com</a>><br>
<b>Sent:</b> Wednesday, August 9, 2023 10:46 PM<br>
<b>To:</b> The Friday Morning Applied Complexity Coffee Group <<a href="mailto:friam@redfish.com" target="_blank">friam@redfish.com</a>><br>
<b>Subject:</b> [EXT] [FRIAM] the Monty Hall problem</font>
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<div>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.
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<p style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:"Calibri",sans-serif">
<b><span style="font-size:12pt;line-height:107%">The Best Explanation of the Solution of the Monty Hall Problem<span></span></span></b></p>
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<span style="font-family:"Times New Roman",serif">Here is the standard version of the Monty Hall Problem, as laid out in Wikipedia:<span></span></span></p>
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<b><i><span style="font-size:10pt;line-height:107%">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?</span></i></b><b><i><span style="font-size:10pt;line-height:107%;font-family:"Times New Roman",serif"><span></span></span></i></b></p>
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This standard presentation of the problem contains some sly “intuition traps”,<a href="#m_-8597096738618129438_m_9217272474254335166_x__ftn1" name="m_-8597096738618129438_m_9217272474254335166_x__ftnref1" title="" id="m_-8597096738618129438m_9217272474254335166OWA30e97a70-120a-e33c-0661-d60a1d90b0b7"><span style="vertical-align:super"><span><span style="vertical-align:super"><span style="font-size:11pt;line-height:107%;font-family:"Calibri",sans-serif">[1]</span></span></span></span></a>
so put aside goats and cars for a moment. Let’s talk about thimbles and peas.<span>
</span>I ask you to close your eyes, and then I put before you three thimbles, one of which hides a pea.<span>
</span>If you choose the one hiding a pea, you get all the gold in China.<span> </span>
Call the three thimbles, 1, 2, and 3. <span></span></p>
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<span><span>1.<span style="font:7pt "Times New Roman""> </span></span></span><span> </span>I ask you to choose one of the thimbles.<span>
</span>You choose 1.<span> </span>What is the probability that you choose the pea.<span>
</span>ANS: 1/3.<span></span></p>
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<span><span>2.<span style="font:7pt "Times New Roman""> </span></span></span>Now, I group the thimbles as follows.<span>
</span>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.<span></span></p>
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<span><span>3.<span style="font:7pt "Times New Roman""> </span></span></span>I ask you to choose whether to
<i>choose from</i> 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
<b><i>group</i></b> should you choose from?<span> </span>ANS: It doesn’t matter.<span>
</span>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.<span>
</span>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:<span>
</span>2/3 x ½ = 1/3.<span> </span><span></span></p>
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<span><span>4.<span style="font:7pt "Times New Roman""> </span></span></span>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.)<span> </span>Should you choose from Group A or Group B?<span>
</span>ANS:<span> </span>Group B.<span> </span>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.<span>
</span>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.<span></span></p>
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<span><span>5.<span style="font:7pt "Times New Roman""> </span></span></span>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.<span>
</span>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.<span> </span><span></span></p>
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<p style="margin:0in;font-size:10pt;font-family:"Calibri",sans-serif">
<a href="#m_-8597096738618129438_m_9217272474254335166_x__ftnref1" name="m_-8597096738618129438_m_9217272474254335166_x__ftn1" title="" id="m_-8597096738618129438m_9217272474254335166OWA41eef8c3-b031-2493-6afc-cae88d88c2e9"><span style="vertical-align:super"><span><span style="vertical-align:super"><span style="font-size:10pt;line-height:107%;font-family:"Calibri",sans-serif">[1]</span></span></span></span></a>
The intuition trap has something to do with the fact that doors, goats, and cars are difficult to group.<span>
</span>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.<span>
</span>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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