4 1. Classical Probability
likely, the n wha t i s th e probabilit y tha t n ha s a fixed point ? Tha t is , wha t
are th e chance s tha t ther e exist s a n i such tha t 7r(i ) = i?
If Ei := {ft(i) = i} denote s th e even t tha t i is a fixed point , the n
(1.6) P(E
ll
n---nElk) = ^ ^,
whenever 1 i\ i& n. Therefore ,
(1.7) £ P^n.-n^-Q^-I .
h-ik
Thanks t o th e inclusion-exclusio n principle ,
(1.8) p l l j ^ )= i _ ^
+
J -
+
. . . + (_i)«+i l
(y*)-4*s+
ni
As n oo , thi s converge s t o X ^ i ( ~ l ) ^ 1 / ^ ' 1 e _ 1 . W e ca n therefor e
conclude that , i n a large rando m permutation , chance s ar e nearl y 1/e that
no fixed point s arise .
2. Conditiona l Probabilit y
A certai n populatio n i s comprise d o f a hundred adults , twent y o f who m ar e
women. Fiftee n o f the wome n an d twent y o f the me n ar e employed . A
statistician ha s selecte d a person a t random fro m thi s population , an d tell s
us tha t th e perso n s o sample d i s employed. Give n thi s information , wha t
are th e chance s tha t a woman wa s sampled ?
If w e wer e no t priv y t o the informatio n give n b y th e statistician , the n
the sampl e spac e woul d hav e N 100 equall y likel y elements ; on e fo r eac h
adult i n th e population . Therefore , P ( W ) = 20/100 = 0.2 , wher e W denote s
the even t tha t a woman i s sampled .
On th e othe r hand , onc e th e statisticia n tell s u s tha t th e sample d perso n
is employed , thi s knowledg e change s ou r origina l sampl e spac e t o a new on e
tha t contain s th e employe d peopl e only . Th e ne w sampl e spac e i s comprise d
of 3 5 peopl e (15 wome n an d 2 0 men) , al l o f who m ar e employe d an d equall y
likely t o b e chosen . Give n th e knowledg e imparte d t o u s b y th e statistician ,
the chance s ar e 15/35 ~ 0.428 5 tha t th e sample d perso n i s a woman .
In general , it is easier t o no t chang e th e sampl e spac e a s ne w informatio n
surfaces, bu t instea d us e th e followin g conditiona l probabilit y formul a o n th e
original sampl e space :
Definitio n 1.9. For an y tw o event s A and B such tha t P(B) 0, th e
conditional probability of A given B i s define d as
(l.») P ( A | B ) : = ^ .
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