I think this might be a more concise explanation: 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.
_______________________________________________________________________ stephen.gue...@simtable.com <stephen.gue...@simtable.com> CEO, https://www.simtable.com <http://www.simtable.com/> 1600 Lena St #D1, Santa Fe, NM 87505 office: (505)995-0206 mobile: (505)577-5828 On Wed, Aug 9, 2023 at 8:46 PM Nicholas Thompson <thompnicks...@gmail.com> wrote: > 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_3313630866437708646__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_3313630866437708646__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.” > -. --- - / ...- .- .-.. .. -.. / -- --- .-. ... . / -.-. --- -.. . > FRIAM Applied Complexity Group listserv > Fridays 9a-12p Friday St. Johns Cafe / Thursdays 9a-12p Zoom > https://bit.ly/virtualfriam > to (un)subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: 5/2017 thru present > https://redfish.com/pipermail/friam_redfish.com/ > 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >
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