1. Discrete Probability 3
Example 1.5 (Th e Boltzman n Statistic) . Suppos e w e wante d t o pu t n
distinct ball s i n N urns , wher e N n. Le t u s assum e tha t th e ball s ar e
thrown a t rando m int o the N urns , s o that al l N
n
possibl e outcome s o f thi s
game ar e equall y likely .
To understand thi s model better, conside r the case N = 4 and n = 3 . W e
then hav e three ball s x, y , an d z, an d fou r urns . Le t A\ : = {xy\ \z\ } . Thi s
notation designate s th e even t tha t ball s x an d y en d u p i n th e firs t urn , i n
this order, an d bal l z land s in the third urn ; th e three vertical lines represen t
the "dividers " tha t separat e th e differen t urns . Not e tha t A\ represent s a
different arrangemen t fro m Ai : = {yx\ \z\ } , bu t th e tw o event s hav e th e
same chanc e o f occurring .
Going bac k t o th e genera l case , le t A* denot e th e even t tha t th e firs t
n urn s en d u p wit h exactl y on e bal l i n eac h (and , therefore , th e remainin g
N n en d u p with n o balls in them). The n th e cardinalit y o f A* is the sam e
as the numbe r o f different way s we can orde r th e n balls ; i.e., n! . Th e latte r
count i s the so-calle d Boltzmann statistic, an d w e have foun d tha t
(!-3)
P
(^*)
=
W-
Example 1.6 (Th e Bose-Einstein Statistic) . Here , the settin g i s very muc h
the sam e a s i t wa s i n Exampl e 1.5, bu t no w th e ball s ar e indistinguishabl e
from on e another . A s a result , th e event s an d A2 o f Exampl e 1.5 ar e
now on e an d th e same . Thus , |A* | = 1 and P(A* ) = l/|fi| , an d i t remain s
to comput e |f2| .
Observe tha t |fi | denote s th e numbe r o f way s tha t w e ca n mi x u p th e
balls together wit h the urn-divider s an d the n la y them dow n al l in a straigh t
line. Becaus e ther e ar e (N - 1) divider s an d n balls , |fi | = {
N+„~l).
Thi s
number i s called th e Bose-Einstein statistic , an d w e have foun d tha t
(!-4)
p
(^*) = -prLy-
Example 1.7 (Th e Fermi-Dira c Statistic) . W e continu e t o wor k withi n
the genera l framewor k o f Example s 1.5 an d 1.6. Bu t now , w e assum e tha t
the ball s ar e al l indistinguishabl e an d eac h ur n contain s a t mos t on e ball .
Evidently, |J1| is equal t o ( ) , th e Fermi-Dirac statistic , an d
O'
Example 1.8 (Rando m Matchings). Conside r n people each of whom drive s
a differen t car . I f w e assig n a ca r t o eac h perso n a t random , the n wha t i s
the probabilit y tha t a t leas t on e perso n get s hi s o r he r ow n car ?
Equivalently, conside r al l permutation s n : { 1 , . . . , n } { 1 , . . . , n};
there ar e n! of them. I f we select a permutation a t random , al l being equall y
(1-5) P(A 0 =
fN
.
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