| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
I made a perfunctory search for this item under several headings and didn't find it, so I'll post it. This problem was posted in the Boston Globe Parade under the Mariliyn vos Savant column a while ago (I don't have the original date) and apparently created quite a controversy, drawing letters both attacking (many) and supporting (few) the supplied solution from many people with mathematical backgrounds. The problem and another defence was posted in the 17-feb edition of the magazine ibid.. Here it is, along with the Globe supplied solution (I've placed the solution after a form feed so as not to spoil for those who want to try it from scratch), but without commentary or detailed explanation. I paraphrase, but I think I've captured all of the essential aspects of the problem. I think this is a good example of how devilish simple sounding problems in probability can get. ------------------------------------------------------------------------ A contestant on a game show is shown 3 doors and is told that behind one is a new car, and behind the other two are goats. The contestant chooses door #1, but before opening the door, the host offers the following help: the host opens door #3 revealing a goat, and allows the contestant a last chance to change his/her choice to door #2. The question is: Should the contestant change his/her choice to door #2 ? The supplied answer is (follows form feed) Yes, the probability that the prize is behind door #2 is 2/3, while that of it being behind door #1 is 1/3.
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1390.1 | see 1078.nn for a discussion of this and a more precisely formulated version | CSSE::NEILSEN | Wally Neilsen-Steinhardt | Mon Feb 18 1991 15:36 | 0 |
| 1390.2 | Answer was wrong, though as good as any answer... | CADSYS::COOPER | Topher Cooper | Mon Feb 18 1991 16:05 | 38 |
The problem, as stated here and in Parade, is indeterminant. Certain
additional assumptions are needed to solve it. As this problem is
usually stated the "host" is specifically Monty Hall and the show is
specifically "Let's Make A Deal". For those who know the show, this
supplies enough information to complete the analysis -- giving
Marilyn's counterintuitive result. This is related both in form and
in philosophy to the problem discussed in 1291.
One obvious assumption that needs to be made is that there *is* one and
only one good prize and the "goat" wasn't it. You also must assume
that the prize is not moved around in response to the contestant's
actions: it either stays in one place or is switched around
independently of the contestant's actions. Similarly to 1291, the most
interesting assumption you must make concerns the host's actions if
circumstances were different than they turned out to be -- in other
words what was the sampling process of which this was one sample of.
The 1/3:2/3 answer comes from the assumption that the host knows where
the "good" prize is and avoids revealing it if the contestant has not
already chosen it. The host's actions therefore reflect information
which the contestant can take advantage of by "switching".
If, on the other hand, the host chooses arbitrarily (i.e.,
independently of the actual location of the "good" prize) than there
is no information about the location reflected by his/her actions and
there is no advantage (or disadvantage) to switching.
More subtly -- if the contestant doesn't know whether or not the host
has deliberately avoided revealing the prize or just chosen randomly
than (assuming that these are the only possibilities) then they should
switch. The advantage will, however, be less than the 2:1 advantage
if they know that the host avoids the prize.
Of course, if the host always reveals the prize unless the contestant
has already revealed it, then the contestant clearly should not switch
if the host reveals a goat.
Topher
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| 1390.3 | ELIS::GARSON | V+F = E+2 | Mon Aug 19 1991 07:41 | 15 | |
re .2
Sorry to wake a sleeping note but a friend of mine (non-DECcie) asked
me about this problem over the weekend and I seemed to recall that it
had already been covered here.
> For those who know the show, this supplies enough information to complete
> the analysis -- giving Marilyn's counterintuitive result.
I'm not familiar with this show. Am I right in assuming that the actual
behaviour of the host is that he does know where the car is and always
chooses to open a door that he knows will not reveal the car? If the
contestant has not picked the car then the host is forced in his choice
of door. If the contestant has picked the car then the host can choose
either of the remaining doors.
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| 1390.4 | ALIEN::EDP | Always mount a scratch monkey. | Mon Aug 19 1991 08:28 | 8 | |
Re .3:
Yes, the correctly-stated problem includes those stipulations: The
host always opens a door not selected by the contestant to reveal a
dud.
-- edp
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| 1390.5 | CLT::TRACE::GILBERT | Ownership Obligates | Tue Aug 20 1991 13:13 | 2 | |
Actually, sometimes the host reveals what appears to be a 'dud', but
when the curtain behind this 'dud' is raised, there is the big prize.
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