A even simpler explanation is that your initial pick has a 1/3 chance of winning. Nothing has changed as far as that door is concerned. Thus, with host removing one of the other two doors, the probability of winning must be 2/3.
These word problems which involve a priori vs a posteriori probabilities can be very tricky. In Contact Bridge, one important variant is known as the principle of restricted choice. Suppose you have 9 cards in a suit but are missing the Q and the J. Suppose you have the A and K in your hand. You play the A and the player on your left drops the Q or J. What do you do next? You can either play the K hoping the other Q or J will drop or you can finesse hoping the player on your right has the missing Q or J. Most people think the two strategies are (approximately) equally probably to succeed but in fact finessing is twice as probable to succeed. It’s really frustrating to be playing a competition and have the opponents play the losing strategy and luck out. Ed __________ Ed Angel Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab) Professor Emeritus of Computer Science, University of New Mexico 1017 Sierra Pinon Santa Fe, NM 87501 505-984-0136 (home) edward.an...@gmail.com 505-453-4944 (cell) http://www.cs.unm.edu/~angel > On Aug 9, 2023, at 9:15 PM, Stephen Guerin <stephen.gue...@simtable.com> > wrote: > > 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 <mailto: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 > <mailto: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] <x-msg://68/#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] <x-msg://68/#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/ > -. --- - / ...- .- .-.. .. -.. / -- --- .-. ... . / -.-. --- -.. . > 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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