Self-assembly and structural transformations of high-nuclearity palladium-rich polyoxometalates†

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1、2Outcomes,events,andprobabilityTheworldaroundusisfullofphenomenaweperceiveasrandomorunpre-dictable.Weaimtomodelthesephenomenaasoutcomesofsomeexperiment,whereyoushouldthinkofexperimentinaverygeneralsense.TheoutcomesareelementsofasamplespaceΩ,andsubsetsofΩarecalledevents.Theeventsw

2、illbeassignedaprobability,anumberbetween0and1thatexpresseshowlikelytheeventistooccur.2.1SamplespacesSamplespacesaresimplysetswhoseelementsdescribetheoutcomesoftheexperimentinwhichweareinterested.Westartwiththemostbasicexperiment:thetossingofacoin.Assumingthatwewillneverseethecoin

3、landonitsrim,therearetwopossibleoutcomes:headsandtails.WethereforetakeasthesamplespaceassociatedwiththisexperimentthesetΩ={H,T}.Inanotherexperimentweaskthenextpersonwemeetonthestreetinwhichmonthherbirthdayfalls.AnobviouschoiceforthesamplespaceisΩ={Jan,Feb,Mar,Apr,May,Jun,Jul,Aug,

4、Sep,Oct,Nov,Dec}.Inathirdexperimentweloadascalemodelforabridgeuptothepointwherethestructurecollapses.Theoutcomeistheloadatwhichthisoccurs.Inreality,onecanonlymeasurewithfiniteaccuracy,e.g.,tofivedecimals,andasamplespacewithjustthosenumberswouldstrictlybeadequate.However,inprinciple

5、,theloaditselfcouldbeanypositivenumberandthereforeΩ=(0,∞)istherightchoice.Eventhoughinrealitytheremayalsobeanupperlimittowhatloadsareconceivable,itisnotnecessaryorpracticaltotrytolimittheoutcomescorrespondingly.142Outcomes,events,andprobabilityInafourthexperiment,wefindonourdoorma

6、tthreeenvelopes,senttousbythreedifferentpersons,andwelookinwhichordertheenvelopeslieontopofeachother.Codingthem1,2,and3,thesamplespacewouldbeΩ={123,132,213,231,312,321}.Quickexercise2.1Ifwereceivedmailfromfourdifferentpersons,howmanyelementswouldthecorrespondingsamplespacehave?Inge

7、neralonemightconsidertheorderinwhichndifferentobjectscanbeplaced.Thisiscalledapermutationofthenobjects.Aswehaveseen,thereare6possiblepermutationsof3objects,and4·6=24of4objects.Whathappensisthatifweaddthenthobject,thenthiscanbeplacedinanyofnpositionsinanyofthepermutationsofn−1objec

8、ts.Thereforetherearen·(n−1)····3·2·1=n!p