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-boun...@redfish.com> on behalf of Nicholas Thompson 
<thompnicks...@gmail.com>
Sent: Wednesday, August 9, 2023 10:46 PM
To: The Friday Morning Applied Complexity Coffee Group <friam@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] 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] 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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