chris@umcp-cs.UUCP (08/24/83)
Everyone (including me) seems to think, "oh, that's easy, it's the
same as the original problem." It's not. The next thought is "well,
then, it's 50%." Wrong again....
It turns out that the probability of a gold coin in the upper drawer
is 75%! This amazing conclusion is reached as follows:
case 1: silver coin of G/S pair is in lower drawer.
In this case, in order to get a gold coin when opening the lower
drawer, you have to get the G/G pair. Thus the upper drawer
contains a gold coin. This means that in case 1, you have a
100% chance of a gold coin.
case 2: silver coin of G/S pair is in upper drawer.
In case 2, you have an equal chance of picking the G/G pair or
the G/S pair. If you pick the G/G pair, you get a gold coin.
If you pick the G/S pair, you get a silver coin. So in case
2, you have a 50% chance.
Case 1 & 2 are equally likely, thus we have:
[chance of case 1] [chance of gold] [total]
0.5 x 1.0 = 0.50
plus
[chance of case 2] [chance of gold]
0.5 x 0.5 = 0.25
----
0.75
or 75% chance.
Judging by the number of wrong answers, this problem is substantially
more difficult than the preceding one. (I'd certainly like to
think so, having twice come up with the wrong answer!)
Chris
--
In-Real-Life: Chris Torek, Univ of MD Comp Sci
UUCP: {seismo,allegra,brl-bmd}!umcp-cs!chris
CSNet: chris@umcp-cs ARPA: chris.umcp-cs@UDel-Relaychris@umcp-cs.UUCP (08/24/83)
Hmm, that's funny; my analysis seems OK but when I run the problem
through the Vax I get . . . 2/3! Oh dear, you seem to have made a
fool of yourself again. *sigh*
Chris ("Well, at least I can get the computer to tell me I screwed up") Torek
--
In-Real-Life: Chris Torek, Univ of MD Comp Sci
UUCP: {seismo,allegra,brl-bmd}!umcp-cs!chris
CSNet: chris@umcp-cs ARPA: chris.umcp-cs@UDel-Relaywdr@security.UUCP (William D Ricker) (08/24/83)
Sorry, Chris, your 75% is wrong.
What you failed to notice is that though case 1 and case 2 are in
general equlally likely, there are not equally likely in the realm of
discussion.
Case 1 and Case 2, Silver down and Silver Up, are equally likely only
before any drawers are opened. They are equally likely set-ups.
However, Two-Thirds of Case One and One-Third of Case two are excluded
from the universe of consideration: the subcases of Silver showing in
the first drawer, a bottom drawer, opened.
Thus in the realm of discussion, a bottom drawer showing gold, Case 2
is TWICE as likely.
I hope this helps clear things up for those of you who had questions
about choosing the state-space to count.
Bill Ricker
(617)271-3725 MS k203, The MITRE Corporation, Bedford, MA, 01730
wdr@security.UUCP (Internet)
{allegra,genrad,ihnp4,utzoo,philabs,uw-beaver}!linus!security!wdr (UUCP)
wdr@mitre-bedford (ARPA)jim@ism780.UUCP (Jim Balter) (08/25/83)
NO! NO! NO! (much screaming and tearing of hair). You have repeated the same old mistake in a new context: Case 1 & 2 are equally likely, thus we have: They are **not** equally likely!!! The stipulation, that you find a G coin when you select a bottom drawer at random, is twice as likely to be satisfied when the S coin is in the top drawer as when it is in the bottom drawer. Therefore, given the stipulation, the odds that the S coin was in fact in the top drawer is twice the odds that it was in fact in the lower drawer (if you don't think this follows, we need a formal linguist; I don't know how to formally demonstrate it). So, the answer is (1/3 * 100%) + (2/3 * 50%) = 2/3 (!!!!). Selecting the bottom drawer every time is just another way of randomly selecting a drawer if the method of placing the coins is random. So the two answers *must* be identical. Substitute "the drawer manufactured earlier" for "the top drawer" and "the drawer manufactured earlier" for "the bottom drawer" and then do the analysis. "But I don't know which was manufactured first", you say? Exactly; it doesn't matter. Jim Balter (decvax!yale-co!ima!jim), Interactive Systems Corp --------