1.SampleSpaceandProbabilityPartV:Coun8ng‐2

ECE302Fall2009TR3‐4:15pmPurdueUniversity,SchoolofECE

Prof.IlyaPollak

Par88ons

•  Howmanywaysaretheretodividenitemsintorgroupssothatthei‐thgrouphasniitems(wheren1+n2+…+nr=n)?

•  Example:ifr=2,theansweris

€

nn1

ways of forming the first group

n − n1 objects remain

n − n1

n2

ways of forming the second group

n − n2 objects remain

€

Answer : nn1

n − n1

n2

⋅… ⋅

n − n1 −…− nr−1

nr

=n!

n1!(n − n1)!⋅

(n − n1)!n2!(n − n1 − n2)!

⋅… ⋅(n − n1 −…− nr−1)!

nr!(n − n1 −…− nr−1 − nr )!= 0!=1

=n!

n1!n2!…nr!

These are denoted n

n1,n2,…,nr

and called multinomial coefficients.

The reason for this name is that

x1 + x2 +…+ xr( )n =n

n1,n2,…,nr

n1 +n2 +…+nr = n∑ x1

n1 x2n2 ⋅… ⋅ xr

nr

Example(p.68)•  90students,includingJoeandJane,aretobesplitinto3

classesofequalsize,atrandom.

•  WhatistheprobabilitythatJoeandJaneendupinthesameclass?

Example•  90students,includingJoeandJane,aretobesplitinto3

classesofequalsize,atrandom.

•  WhatistheprobabilitythatJoeandJaneendupinthesameclass?

Example•  90students,includingJoeandJane,aretobesplitinto3

classesofequalsize,atrandom.

•  WhatistheprobabilitythatJoeandJaneendupinthesameclass?

Example•  90students,includingJoeandJane,aretobesplitinto3

classesofequalsize,atrandom.

•  WhatistheprobabilitythatJoeandJaneendupinthesameclass?

Asimplermethod

•  PlaceJoeinoneclass.

Asimplermethod

•  PlaceJoeinoneclass.•  ForJane,thereare89totalpossibili8es,ofwhich29putherinthesameclasswithJoe.

Asimplermethod

•  PlaceJoeinoneclass.•  ForJane,thereare89totalpossibili8es,ofwhich29putherinthesameclasswithJoe.

•  Answer:29/89.