4 1. M o d e l i n g a Probabilisti c E x p e r i m e n t
In particular , w e hav e tha t P({UJ 1 }) = p
2
, whic h w e wil l writ e simpl y
as P(UJ 1= ) p
z
.
We le t \
A
b e th e characteristic function o f A] tha t is , \
A
i s th e
function mappin g ft t o {0,1} tha t take s th e valu e 1 o n A an d th e
value 0 o n it s complemen t A c. Thu s
d
P(A) = ^XA^')-
In summary , ou r mathematica l mode l i s define d b y a pai r (ft , P )
where ft i s a finite se t an d P i s a functio n fro m th e se t o f subset s o f
ft t o th e interva l [0,1] satisfyin g th e followin g tw o conditions :
(1) P(ft ) = l .
(2) I f A an d B ar e disjoin t subset s o f ft, the n P(A U B) =
P(A) + P{B).
A pai r (ft , P) satisfyin g thes e condition s i s calle d a finite probability
space an d th e functio n P i s calle d a probability. I t i s eas y t o chec k
the followin g propertie s o f P:
(1) P(0 ) = O .
(2) If ACQ, the n P ( / l c ) = 1 - P ( A ) .
(3) If A, Be ft, the n P ( A U P) = P ( A ) + P ( P ) - P ( A n B).
In th e specia l cas e wher e al l th e outcome s ar e equall y likely , w e
say tha t th e spac e ft ha s a uniform probability. I n thi s case , i t i s eas y
to calculat e th e probabilit y o f a n event : thi s probabilit y i s simpl y
the numbe r o f element s i n th e even t divide d b y d , th e numbe r o f
elements o f ft. Thi s situatio n i s describe d b y th e well-know n rul e tha t
"the probabilit y o f a n even t i s th e rati o o f th e numbe r o f favorabl e
outcomes t o th e tota l numbe r o f possibl e outcomes" .
Here ar e a fe w examples .
E x a m p l e . Th e flip o f a fai r coi n i s describe d b y a se t ft o f tw o
elements an d a probabilit y givin g th e sam e valu e t o eac h o f th e tw o
outcomes. I f w e le t 1 represen t th e outcom e heads an d 0 represen t
the outcom e tails, the n ft = {0,1} an d P(0 ) = P ( l ) = \. W e sa y
tha t th e spac e {0,1} i s equippe d wit h th e unifor m probabilit y (^ , | ) .
Previous Page Next Page