Erin (others in the thread, see below),
Erin Noonan <enoonan@kent.edu> wrote:
>GLENN: When you choose door 57, you have to admit that you probably have it
>wrong.>Nothing Monty subsequently reveals to you will change that, because you
>>made the choice when you had less information. Monty then
>>opens 98 doors showing booby prizes, purposely avoiding revealing the
>>one with the good prize. So when only doors 57 and 66 remain, you can be
>>nearly assured that 66 is the winner.
>
>
>ERIN: ooohhh now I see your reasoning and it makes sense. But I
>am still curious to how the probability is calculated.
>So you are saying these events are not independent like the coin
>toss....so how do you calculate the probability of nonindependent
>events.
You asked for it. Take a deeeeep breath....:
In the 100 door version of the puzzle, your initial guess has a probability
of 1/100 to have the good prize. All 100 doors have this probability, and
the sum of their probabilities add to 1 (as they must). When Monty opens a
door that shows a booby prize, the probability that that door has the good
prize drops to 0 (obviously). Now unless the probability changes for some
of the unyet open doors, the sum of the probabilities doesn't equal 1
anymore (it would equal 99/100), and that is a no-no. Since the probability
of your original pick is fixed at 1/100, the other 99/100 of the
probability has to be spread evenly across the remaining OTHER unopened
doors. Since there are 98 other unopened doors, the probability of each
containing the prize is 99/9800 (99/100 divided by 98). This is a smidge
better than 1/100, which is what they used to be.
As each door is opened, the probability that the good prize is in one of
the as yet unopened doors (except your original pick, which stays fixed)
increases. When he has opened 98 doors, all that stands closed in front of
you is your original pick and one other door. Since the sum of the
probabilities of these two must equal 1, and the probability of your original
pick is still 1/100, the other door must have probability 99/100.
Said another way, you have a 99% chance of picking the good prize of you
switch your pick to the other door.
Lawry, Andrea, Jonathan M, John B,
Your statements about probabilities of coin-flips are correct, but
misapplied here. Please re-think the problem.
If Monty was as clueless as the contestant about what was behind the doors,
and he happened by luck to open a door showing a booby prize, then I would
agree that there is no advantage to switching. But he isn't clueless and
this makes all the difference. I stand firm to my answer.
Glenn
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