2148:
the limit of the ratio between (the number of times you throw the white dice and it shows four or five) and (the number of times you throw the white dice altogether). The conditional probability p(A|B) is the limit of the ratio between (the number of times that you throw both dice and the white dice show four or five and the blue dice show six) and (the number of times that you throw both dice and the blue dice show six). Now, if the two events are independent in the non-technical sense of the word, that the result of the two dice do not depend on one another, then the two limits must be equal. So p(A)=p(A|B) if A and B are independent. Now define that A and B are
2153:
times you throw both dice and the blue one shows six), multiplied by the limit of the ratio between (the number of times the blue dice shows six) and (the number of times you throw the blue dice all together). Using that the product of limits is the limit of the product, you get the limit of the ratio between (the number of times you throw both dice and the white one shows four or five and the blue one shows six) and (the number of times you throw the blue dice all together), which is equal to the right hand side p(A and B).
2147:
No empirical observations are needed, (nor are they possible because the probability is a limit which is not accesible observationally), but a thought experiment: Consider an event A, (say, that a white dice show four or five), and an event B, (say, that a blue dice show six). The probability p(A) is
1013:
This answer got me thinking about whether the
Fundamental Theorem is actually necessary here. More precisely: firstly, does there exist an integral domain in which any pair of elements has a GCD and an LCM, but which is not a UFD? (edit:yes) Secondly, if there are such rings, does this result hold in
2274:
On a certain bike there are spokes that are 14 inches long. Each spoke forms an angle of 30 with each of the two spokes beside it. What is the distance between the places where two spokes that are beside each other attach to the wheel? I think this has something to do with geometric mean, but I'm
2236:
Yes, the 'probability of an outcome of an experiment', based on some hypothesis, is a frequentist probability. The
Bayesian point of view is the opposite one, to estimate the credibility of a hypothesis, based on given outcomes of experiments. While the two interpretations of probability differ, the
871:
Ahaha. Well, that I can live with (although I do often rail against the mathematical tradition of naming things after people instead of descriptively). But Lang is always doing things like assuming "normed instead of metric" so that "we can write the distance in terms of the absolute value sign",
1528:
Hi. First off, no you can't say mutually exclusive events are a special case of independent events, as by definition the two are not independent - if you have one, you cannot have the other. The proof of independence I admit I can't remember right now, but it is fairly logical when you think about
2152:
if p(A)=p(A|B). So independent events are also statistically independent. Consider the equation p(A|B)·p(B)=p(A and B). The left hand side is the limit of the ratio between (the number of times you throw both dice and the white one shows four or five and the blue one shows six) and (the number of
1429:
I think that's what you'd already worked out. It is, indeed, curious that every pair having a GCD implies every pair has an LCM, but a given pair having a GCD doesn't imply that that same pair has a LCM. (The book I found was very large, very old and very much falling apart, so I left it in the
881:
The problem with descriptive naming is that disambiguation can make things pretty unwieldy. My own preference is a combination: thus Hahn-Banach extension theorem, Tietze-Urysohn extension theorem, Carathéodory's extension theorem, etc. On the subject of weak hypotheses in exercises, this often
2320:
Hmm, I wonder why that tripped me up so badly. Is there any way to do this with triangles and trig? This was on a trig test so I have no idea why there would be something about circles on it. I like
Mattbuck's method though. That's a lot simpler than what I was trying. Thanks,
1617:, so there is nothing to prove. You may ask, why did we choose to call "independent" two events satisfying this. I think this boils down to the empirical observation that real-world events which seem independent to us tend to satisfy this rule. You can rephrase this in terms of
2180:
But again, you just move the problem a little further along. In order to justify that two dice will be independent of each other, you first assume that successive tosses of a single die will be independent, with that assumption and its definition swept under the carpet.
1424:
I was unable to find any books which mentioned GCD domains, I did however find one that briefly mentioned a "ring with a greatest common divisor" - it seems the name is far from standard. The only relevant thing it mentioned about them was a theorem:
1502:
my doubt is that ...how can we represent two indepentent events on a venn diagram.if we do it by two intersecting circle .then what can we say about the condition of independence.p(A intersection B)=p(A)*p(B).how is this derived.
1860:
Which, as Meni said, just move the problem to the definition of conditional probability. Sooner or later you just have to accept that those definitions seem to work - you can't prove everything, you have to start somewhere.
1217:
That we're in a GCD domain is only assuming the existence of GCDs, not LCMs, as far as I can tell, so you need a slightly stronger assumption than just being in a GCD domain. (It may turn out to be equivalent, of course.)
872:
or adding hypotheses because "this is the only case anyone cares about anyway." I suspect this is just another combination of his perverse sense of humour and love of dropping stealth exercises for the reader.
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it. If you have two events, neither of which influences the other, the probability of them both happening would be the probability of one happening multiplied by the probability of the other happening. -
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The very concept of the probability of an outcome of an experiment assumes that the experiment can be repeated indefinitely and that the outcome of the repetitions are mutually independent.
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The book I was reading could easily have pre-dated 1974. Odd that none of the other books I looked at mentioned that name, though - maybe Durham Uni library is very out-of-date! --
2335:
Mattbuck's answer gives the distance along the wheel, i.e. along an arc of a circle. If you want the straight-line distance, then trigonometry is the way to go, specifically the
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Oh, I understand what's going on now. I just completely misunderstood the problem. Otherwise, I would've hadn't any trouble solving it. Thank you all for your help,
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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
2351:
Wait, crap. I feel stupid now. I misunderstood the problem and took it that the spokes didn't touch at any point. Well, that was a stupid mistake.
2220:
is a little different. I don't think that's an issue for this discussion, where such frequencies are meaningful, but it's always good to be precise. --
1388:
My algebra's been slow lately, but finding a piece of paper has finally allowed me to prove that any GCD domain has (binary) LCMs. Unfortunately,
957:
Yeah. For LCTVSs you can prove the continuous-extension version of HB from the most basic version via messing around with
Minkowski functionals.
1232:
Sorry, by we I mean me, in my initial question above. Our article doesn't state whether a GCD domain automatically has LCMs, and it should.
37:
1392:
is completely unreferenced and my algebra textbook doesn't mention the things explicitly; time to go looking for a reference that does.
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2298:. You then use the formula for the circumference (c = 2(pi)r), but you only want 30 degrees rather than 360, so multiply by 30/360. -
1448:'s textbook in 1974, so it may now be standard. Other terms mentioned are 'pseudo-Bezout', 'HCF-ring', 'complete' and 'property BA'.
1415:
Google books gave me a ref, which I've added. Curiously, in any ID, ({x,y} has an LCM) → ({x,y} has a GCD), but the converse fails.
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Hmm, how would you pull that off? I tried that on the test, but I couldn't figure out how to make it work. Please explain.
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hey ive got fundamental dobts...kindly some one give me a link or the required answers clearly... i wd b highly greatful ..
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1323:) and vice versa - as per my argument above (with a little tightening up to allow for associates). And if further
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999:), and then work out what the lcm and gcd are in terms of those primes, and it should follow from that. --
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Well, it works, though the use of the basis B is completely unnecessary. You could just end by saying 'if
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An integral domain with a greatest common divisor is one with a least common multiple, and conversely.
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occured or not. This definition seems intuitively natural. As the conditional probability satisfies
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Oh, that's interesting. The name of the theorem confuses me at times. I should know by now that
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algebra is the same. Computing the probability of the outcome based on the hypothesis is
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And of course we're assuming the existence of LCMs, so any minimal CM is an LCM. Thanks.
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2243:. Computing the credibility of the hypothesis based on the observations is called
980:( where lcm stands for least common multiple and gcd for greatest common divisor )
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a finite-dimensional subspace. The claim is that there exists a closed subspace
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I'm off to the Uni library in a bit anyway - I'll see if I can find anything. --
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This is in fact not a homework. It was a bonus question on a test we had today
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can we say mutually exclusive events as a special case of independent events.
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2131:? And, why would you define the latter without empirically observing it? --
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product of two natural numbers is equal to the product of their lcm and gcd
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I am sure this must be in a standard text somewhere - I will add it to the
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serves the purpose of making the reader think more. In some cases (the
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To the best of my memory, it is also true for locally convex spaces.
1170:(and vice versa, by reversing the above), even if it is not unique.
568:{\displaystyle \beta -\sum _{i}\varepsilon _{i}(\beta )e_{i}\in G,}
1972:" (I could quibble about what happens when the probabilities of
995:
I would start by expressing a and b as products of primes (see
2294:
Assuming they come from the same point, you can treat them as
270:{\displaystyle \{\varepsilon _{1},\ldots ,\varepsilon _{n}\}}
974:
could please give me the proof for the following equality
79:
Welcome to the
Knowledge Mathematics Reference Desk Archives
1635:
The definition of statistical independence can be written
1185:
I suspect you might be right, but your proof doesn't work.
1980:
are zero, but never mind that). But how do you know that
1829:, illustrating that the area of the intersection between
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According to my source, 'GCD domain' was popularised by
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springs to mind), the minimal hypotheses are the proof.
744:{\displaystyle v-\sum _{i}\varepsilon _{i}(v)e_{i}\in G}
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494:{\displaystyle \beta \in B-\{e_{1},\ldots ,e_{n}\}}
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2124:{\displaystyle p(A|B)={\frac {P(A\cap B)}{P(B)}}}
2053:, unless you observe that empirically or define
1663:is the same as the unconditional probability of
1655:), meaning that the conditional probability of
1014:them, i.e. is ab always an associate of (a,b)?
1430:library, so don't ask any more questions!) --
360:extends to a continuous linear functional on
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2046:{\displaystyle p(A|B)\cdot p(B)=p(A\cap B)}
1745:{\displaystyle p(A|B)\cdot p(B)=p(A\cap B)}
1339:is not just a maximal c.d. but is a GCD of
104:Complements of finite-dimensional subspaces
2275:not sure and have no idea how to do it. *
364:. The intersection of the kernels of the
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1374:article when I have found a reference.
777:is complete was unused. Dispose of it.
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322:{\displaystyle \{e_{1},\ldots ,e_{n}\}}
206:{\displaystyle \{e_{1},\ldots ,e_{n}\}}
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1817:is derived. In a unit square diagram,
1319:that are also c.m.s are associates of
1291:that are also c.d.s are associates of
1195:common multiple, but not that it is a
802:is cavalier with his hypotheses. ;)
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1821:may be drawn as a horizontal bar and
1263:Aren't we going in circles here ? If
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1610:{\displaystyle p(A\cap B)=p(A)p(B)}
1078:must be a LCM because if we had a
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997:Fundamental theorem of arithmetic
1307:is a minimal common multiple of
923:Right. Stealth exercises. ;)
811:Unfortunately, doing functional
384:{\displaystyle \varepsilon _{i}}
353:{\displaystyle \varepsilon _{i}}
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1038:Seems to me that once you have
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647:Is this the right approach?
603:{\displaystyle \beta \in F+G}
430:{\displaystyle F\cap G=\{0\}}
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1287:(i.e. the only multiples of
773:Oh, and the assumption that
151:{\displaystyle E=F\oplus G.}
2212:As I understand it, that's
1965:{\displaystyle P(A|B)=P(A)}
1825:as a vertical bar crossing
1315:(i.e. the only divisors of
1150:) that is also multiple of
217:and extend this to a basis
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2270:Odd trig/geometry question
1912:Of course you can define "
1158:is a GCD. So for each GCD
391:is then a closed subspace
2150:statistically independent
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1667:. So the probability of
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18:Knowledge:Reference desk
1619:Conditional probability
1110:) is common divisor of
1090:and also a divisor of
970:proof for a.b=lcm x gcd
637:{\displaystyle F+G=E\,}
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764:Good point. Thanks!
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1623:Meni Rosenfeld
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840:Stefan Banach
838:
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2183:Black Carrot
2149:
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1973:
1917:
1913:
1894:
1890:
1886:
1882:
1878:
1842:
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1672:
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1553:independence
1548:
1530:
1505:
1501:
1498:
1495:set theories
1426:
1364:
1360:
1359:is a LCM of
1356:
1352:
1348:
1344:
1340:
1336:
1328:
1324:
1320:
1316:
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1675:on whether
1673:independent
1516:—Preceding
26:Mathematics
2341:Algebraist
1549:definition
1450:Algebraist
1417:Algebraist
1394:Algebraist
1390:GCD domain
1372:GCD domain
1333:GCD domain
1234:Algebraist
1210:Algebraist
1201:Algebraist
1033:GCD domain
1016:Algebraist
959:Algebraist
888:Algebraist
884:five lemma
844:Algebraist
779:Algebraist
753:Algebraist
437:. And if
329:. By the
120:such that
2251:Bo Jacoby
2199:Bo Jacoby
2155:Bo Jacoby
1899:Bo Jacoby
1847:Bo Jacoby
1376:Gandalf61
1331:are in a
1172:Gandalf61
1118:(because
50:<<
2301:mattbuck
1532:mattbuck
1335:so that
982:Kasiraoj
816:requires
813:analysis
578:so that
24: |
22:Archives
20: |
1518:comment
1295:) then
1193:minimal
1070:and of
925:— merge
874:— merge
804:— merge
766:— merge
649:— merge
333:, each
225:. Let
89:pages.
2415:Zrs 12
2353:Zrs 12
2323:Zrs 12
2281:Zrs 12
1659:given
831:things
99:May 21
67:May 22
46:May 20
2383:. --
2296:radii
2222:Tango
1863:Tango
1459:Tango
1432:Tango
1403:Tango
1347:then
1220:Tango
1197:least
1191:is a
1094:then
1001:Tango
837:after
834:named
683:then
501:then
69:: -->
63:: -->
62:: -->
44:<
16:<
2419:talk
2389:talk
2357:talk
2327:talk
2307:Talk
2285:talk
2255:talk
2226:talk
2203:talk
2187:talk
2159:talk
2137:talk
1916:and
1903:talk
1889:) =
1867:talk
1851:talk
1841:and
1833:and
1647:) =
1627:talk
1547:The
1538:Talk
1512:talk
1463:talk
1436:talk
1407:talk
1380:talk
1363:and
1343:and
1327:and
1311:and
1283:and
1275:and
1267:and
1224:talk
1176:talk
1134:and
1114:and
1086:and
1050:and
1042:GCD
1005:talk
986:talk
950:twma
800:Lang
221:for
161:Let
108:Let
1976:or
1671:is
1555:is
1551:of
1514:)
1046:of
825:lot
751:.'
395:of
60:Jun
56:May
52:Apr
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2096:∩
2035:∩
2008:⋅
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1857:.
1853:)
1845:.
1793:⋅
1769:∩
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842:.
828:of
736:∈
708:ε
698:∑
694:−
668:∈
644:.
589:∈
586:β
557:∈
541:β
529:ε
519:∑
515:−
512:β
473:…
454:−
448:∈
445:β
410:∩
373:ε
342:ε
301:…
256:ε
249:…
237:ε
185:…
140:⊕
58:|
54:|
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2113:B
2110:(
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2099:B
2093:A
2090:(
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2081:=
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2075:B
2071:|
2067:A
2064:(
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2041:)
2038:B
2032:A
2029:(
2026:p
2023:=
2020:)
2017:B
2014:(
2011:p
2005:)
2002:B
1998:|
1994:A
1991:(
1988:p
1978:B
1974:A
1960:)
1957:A
1954:(
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1948:=
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1934:A
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1918:B
1914:A
1901:(
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1799:(
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134:=
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118:G
114:F
110:E
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