Subject:
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Re: math question (or pattern... whatever...)
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Newsgroups:
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lugnet.off-topic.geek
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Date:
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Tue, 4 Mar 2003 15:37:26 GMT
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Viewed:
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325 times
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In lugnet.off-topic.geek, Bruce Schlickbernd writes:
> In lugnet.off-topic.geek, David Eaton writes:
> > Actually, there's a 2/3 chance that it's behind door Z! A very sneaky
> > problem, in fact. It was shown to us in high school as the subject of a very
> > minor controversy. After being published as "2/3 chance" in a math
> > publication, some college math professor (and his class) wrote in and said
> > that it was in fact 1/2; which in turn inspired the magazine to further
> > explain the answer :)
>
> Inasmuch as you get to pick a second time, the odds are 50/50. If you did
> not get to pick a second time, the odds would be 1/3, not 1/2.
For any still not getting this one (it's tricky!) here's another way to
analyze the problem. List all outcomes:
Prize Choice Door Revealed Stay or Switch?
1 1 2 stay!
1 2 3 switch!
1 3 2 switch!
2 1 3 switch!
2 2 1 stay!
2 3 1 switch!
3 1 2 switch!
3 2 1 switch!
3 3 1 stay!
Notice that only 1/3 of the time would you win by sticking with your
original door. The other 2/3 of the time, you win by switching.
> Of course,
> this is leaving out the poker aspect: if your initial pick was incorrect,
> would they even offer the second chance? In fact, they did, but what
> percentage of the time I couldn't tell you, but I would suspect that they
> decided on showing a wrong door no matter what as part of the script (take X
> long to make a presentation).
Note above that I don't list both:
Prize Choice Door Revealed Stay or Switch?
1 1 2 stay!
1 1 3 stay!
Why? Because you're entirely right. The game show host has an algorithm for
picking the door(s) he's going to reveal to you before giving you the option
to switch. He'll *NEVER* show you the door you picked, and he'll *ALWAYS*
show you a goat. (Also note that if I had listed the above 2 choices,
there'd be 2 entries for Prize 1, Choice 1, but only *1* entry for (say)
Prize 2, Choice 1)
> They can *always* show you a door that the car is not behind. It doesn't
> improve your chances at all if there is no second chance to pick. What you
> are saying is that it is *always* a 2/3 chance that you are right.
Actually, there's a 1/3 chance that *YOU* are right because you picked the
right door. But whichever door the *HOST* picks as the final door for you to
guess between is different, because the host has secret knowledge of which
door the prize is behind. 1/3 of the time, you'll pick the correct door, and
the host will be *forced* to give you a "final choice door" that's
incorrect. But the other 2/3 of the time, your choice is wrong, and the door
the host gives you as the "final choice door" is guaranteed to be the
almighty prize-winning door.
It's a tough problem-- if you still need more convincing, try this with a
friend:
Have your friend pick a number between 1 and a million, and write it on a
piece of paper. Then you try to guess the number. Your friend now says, ok,
it's either the number you picked, OR it's <another number>. Try switching
to whatever number your friend gives you as a final choice *EVERY* time, and
see what happens.
DaveE
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Message has 2 Replies:
 | | Re: math question (or pattern... whatever...)
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| (...) I think your analysis is incorrect, precisely because you did *not* list all outcomes. Choose Prize Door Revealed Possible choices 1 1 2 Stay (win). Switch (lose). 1 1 3 Stay (win). Switch (lose). 1 2 3 Stay (lose). Switch (win). 1 3 2 Stay (...) (22 years ago, 4-Mar-03, to lugnet.off-topic.geek)
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Message is in Reply To:
| | Re: math question (or pattern... whatever...)
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| (...) Inasmuch as you get to pick a second time, the odds are 50/50. If you did not get to pick a second time, the odds would be 1/3, not 1/2. Of course, this is leaving out the poker aspect: if your initial pick was incorrect, would they even offer (...) (22 years ago, 4-Mar-03, to lugnet.off-topic.geek)
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