- #36
StoneTemplePython
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r0bHadz said:I understand bayes rules and I understand why you have a 1/3 chance if you remain with your door. And I understand that there has to be another 2/3 somewhere to satisfy 1/3 + x = 1. What I don't understand is how does monty revealing one of the doors give the remaining door that you have the option to switch to, give it a probability of 2/3, since it started off with a probability of 1/3 before he revealed anything.
do you want symbol manipulation or a heuristic / intuitive argument? My feeling is that page 1 has more than enough for intuition. Note: Erdos was finally convinced by a simulation -- there's no shame in getting your intuition from simulations.
As far as symbol manipulation goes, I've given the most accessible one I can think of...
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perhaps for intuition:
(i) either just accept it is the fact of how complementary events work (not totally satisfying).
(ii) alternatively, and very roughly, you could think of it as 1/3 of the time you are right to begin with so Monty's maneuvers are irrelevant... but 2/3 of the time you have chose wrongly and Monty is 'merging' all of the other options into one door for you. I.e. 2/3 of the time Monty is doing you a huge favor by narrowing the options down to one door that is correct, and 1/3 of the time what Monty is doing is irrelevant. (Ok this is really just a rehash of (i), I suppose).
From a symbol manipulation standpoint, you could try applying Bayes Rule to some door you didn't select, but it'll be a bit messy. There's something of a reference frame problem underneath this -- the frame of reference is the door you selected and its hard to disentangle from that.
I suspect this is what is inhibiting intuition here but I won't dwell on this. My view is sometimes you need to find (at least) 2 approaches to a problem -- one to get the proof and another complementary one (ha) to reinforce your intuition.
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