Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 8

From Fig. 1.6 , it follows:

PfAg¼PfA \ BgþPfA \ Bg; (1.27)

PfBg¼PfA \ BgþPfA \ Bg: (1.28)

Adding and subtracting P { A\B }in( 1.26 )-and also using ( 1.27 ) and ( 1.28 )-we
arrive at:

PfA [ Bg¼PfA \ BgþPfA \ BgþPfA \ BgþPfA \ BgPfA \ Bg

¼½PfA \ BgþPfA \ Bgþ½PfA \ BgþPfA \ Bg PfA \ Bg

¼ PfAgþPfBgPfA \ Bg:

(1.29)

1.3 Equally Likely Outcomes in the Sample Space

Consider a random experiment whose sample space S is given as:

S ¼fs 1 ; s 2 ; ...; s N g: (1.30)

In many practical cases, it is natural to suppose that all outcomes are equally
likely to occur:

Pfs 1 g¼Pfs 2 g¼¼Pfs N g: (1.31)

From Axioms II and III, we have:

PfSg¼ 1 ¼ Pfs 1 gþPfs 2 gþþPfs N g; (1.32)

which implies, taking ( 1.31 ) into account:

Pfs 1 g¼Pfs 2 g¼¼Pfs N g¼ 1 =N: (1.33)

From ( 1.33 ) and Axiom III it follows that for any A , as defined in S , the following
will be true:

Number of outcomes in A
Number of outcomes in S :

PfAg¼ (1.34)

Example 1.3.1 Consider the outcomes of the sample space S for the die rolling
experiment. It

is natural to consider that all outcomes are equally probable,

resulting in:

Pfs i g¼ 1 = 6 ; for i ¼ 1 ; ...; 6 : (1.35)