4 ms·
This has already been explained elsewhere in this thread, but here it goes. The original problem formulation in Parade magazine is as follows: Suppose you're o
by em500 4y ago
This has already been explained elsewhere in this thread, but here it goes. The original problem formulation in Parade magazine is as follows:
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?
Assumptions about the strategy or decision rule of the host are not irrelevant, but crucial. Suppose the hosts' decision rule (unknown to the candidate) is
IF (candidate's initial choice is correct):
open one of the remaining goat doors
ELSE:
don't open anything
then it's of course not advantagous to switch. The problem with the original Parade formulation is that it only describes what the candidate sees and what the host actually did in one particular scenario, not what rules the hosts must obay. So you don't know if the hosts is playing adversarial games, but you can't rule it out either.
That's why (as per linked article) in later formulations the "standard assumptions" (that the hosts will always open a door with a goat) are usually given explicitly (in which case the solution is also a lot more obvious).
- Jtsummers 4y agoIt's entirely possible the host is a cheat and the prize is always moved to whichever door you didn't select so you only can ever get a goat. So don't play the game. We can come up with a million what-ifs, but none have a basis in the problem statement so there's no point in them.
- em500 4y agoThe original Parade problem is fairly ambiguous and incomplete. Therefor there is no "correct solution". Other formulations typically include the conditions that the host will always open a door, and that door will always contain a goat as explicit assumptions. In that case the math and logic are pretty straightforward. But in the setting of an actual repeated game show, I think the more realistic case to keep the show interesting is that the host will mix up his strategy, i.e. sometimes he opens another door, sometimes he doesn't. If it's a long running daily or weekly show, if he always opens a door with a goat, the public will work out pretty quickly that it's advantageous to always switch.
- mcphage 4y agoThe original formulation was probably written under the assumption that the readers were familiar with the game show it was drawn from.
- em500 4y agoAs TFA notes the Parade magazine formulation is only "loosely based" on the actual show Lets Make a Deal" hosted by Monty Hall. The real Lets Make a Deal* is a bit more complicated[1], and AFAICT there is no easy analysis or solution for an optimal strategy. The game described in Parade magazine is different from the TV show game, and the assumptions (about the host's behavior) are incomplete/ambiguous enough that switching is not necesarrily the optimal strategy. This, like almost everything else you can think about the Monty Hall Problem, is also described in TFA[2]. [1] https://en.wikipedia.org/wiki/Let%27s_Make_a_Deal#Format https://en.wikipedia.org/wiki/Let%27s_Make_a_Deal#Format [2] https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...