<div dir="ltr">I think this might be a more concise explanation:<br><br>Switching wins if you initially pick a goat (2/3 chance) and loses if you pick the car (1/3 chance), so the win probability with switching is 2/3.<br><br clear="all"><div><div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature"><div dir="ltr"><div dir="ltr"><div dir="ltr">_______________________________________________________________________<br><a href="mailto:stephen.guerin@simtable.com" target="_blank">Stephen.Guerin@Simtable.com</a><div>CEO, <a href="http://www.simtable.com/" target="_blank">https://www.simtable.com</a><br><div>1600 Lena St #D1, Santa Fe, NM 87505<div><div>office: (505)995-0206 <span style="font-size:12.8px">mobile: (505)577-5828</span></div><div></div></div></div></div></div></div></div></div></div><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Wed, Aug 9, 2023 at 8:46 PM Nicholas Thompson <<a href="mailto:thompnickson2@gmail.com">thompnickson2@gmail.com</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 dir="ltr"><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. <br></div><div><br></div><div>
<p class="MsoNormal" 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>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><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>
<p class="MsoNormal" style="margin:0in 0in 8pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><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>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">This standard presentation of the problem contains some sly
“intuition traps”,<a href="#m_3313630866437708646__ftn1" name="m_3313630866437708646__ftnref1" title=""><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>
<p style="margin:0in 0in 0in 0.25in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><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>
<p style="margin:0in 0in 0in 0.25in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><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>
<p style="margin:0in 0in 0in 0.25in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><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>
<p style="margin:0in 0in 0in 0.25in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><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>
<p style="margin:0in 0in 8pt 0.25in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><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>
<div><br clear="all">
<hr width="33%" size="1" align="left">
<div id="m_3313630866437708646gmail-ftn1">
<p style="margin:0in;font-size:10pt;font-family:Calibri,sans-serif"><a href="#m_3313630866437708646__ftnref1" name="m_3313630866437708646__ftn1" title=""><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.” <span></span></p>
</div>
</div>
</div></div>
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