2 1. Classical Probability
Definitio n 1.1 . W e cal l ft := {wj}JLi ^ n e sample space o f ou r experiment .
Subsets o f ft ar e calle d events, an d th e probability o f a n even t A i s
(1.1) P(^): = J2 Pi-
i i : ujeA
One ca n chec k directl y tha t P ha s th e followin g properties :
P r o p o s i t i o n 1.2. P(ft ) = 1, P ( 0 ) = 0 , and
(1) P ( 4 i U A2) = P(Ai ) + P(A
2
) - P(A i n A
2
).
(2) P(,4 C) = 1 - P ( , 4 ) .
(3) If the Ai 's are pairwise disjoint, then P(U^
1
-Aj) = Yl^Zi P(-^i) -
(4) If A
x
C A
2
, toen P(Ai ) P(A
2
) .
(5) (Boole' s Inequality ) P ( U ~ 1. 4
i
) E £ i P ( ^ i ) -
(6) (Inclusion-Exclusio n Formula )
p(U^)=Ep(^)-
E E
p
(^
n
4)
+ -- - + ( - l f - 1 P ( y l
1
n - - - n ^
n
) .
The cas e wher e ft {u)i\f=1 an d pi = = PN 1/^V deserve s specia l
mention. I n thi s case , th e cjj' s ar e equally likely t o occur , an d
(1.2)
P
M = js} = ^
for al l event s A , wher e | | denote s cardinality .
E x a m p l e 1.3. I f w e ar e tossin g a coi n (th e experiment) , the n w e ca n thin k
of ft a s {H,T}, wher e H stand s fo r "heads " an d T fo r "tails. " If th e coi n
is fair , the n p\ an d P2 ar e bot h equa l t o 1/2, an d th e precedin g definitio n
states that : (a ) P ( 0 ) = 0 ; (b ) P({H}) = 1/2; (c ) P({T} ) = 1/2; an d (d )
P ( n ) = P ( { f T , r } ) = (l/2 ) + ( l / 2 ) = l .
E x a m p l e 1.4. Conside r th e se t ft : = {(1,1), ( 1 , 2 ) , . . ., ( 6 , 6)}; thi s coul d
denote th e sampl e spac e o f a rol l o f a pai r o f fai r dice . Fo r example , (i , j)
may correspon d t o i dot s o n th e firs t di e an d j o n th e second . Then , uj\ =
( 1 , l),u; 2 = ( 1 , 2 ) , . . . ar e equall y likely , an d eac h ha s probabilit y 1/36 be -
cause |ft | = 36 . Th e collectio n A := {(1,1), ( 2 , 2), (3 , 3), (4 , 4), (5 , 5), (6 , 6)}
is th e even t tha t w e rol l doubles ; i t ha s probabilit y P(A) = 1/6.
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