Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 15

Then,

PfAg¼ X

N

PfAjB i gPfB i g: (1.65)

i¼ 1

Proof Event A can be written as:

A¼ðA\B 1 Þ[ðA \B 2 Þ[ðA \B 3 Þ[ðA\B 4 Þ[ðA\B 5 Þ¼ [

N
i¼ 1 ðA \B i Þ:

(1.66)

From here, using ( 1.13 ) and ( 1.43 ) we arrive at:

( )

PfAg¼P [ ¼ X PfA \ B i g¼ X

N
i¼ 1 ðA \ B i Þ

N N

PfAjB i ÞPðB i Þ: (1.67)

i¼ 1 i¼ 1

Example 1.6.2 The theorem of total probability is illustrated in Fig. 1.10 for N ¼ 5.
Events B 1 ,

, B 5 are mutually exclusive. Event A which is the union of N mutually
exclusive events A\B i , i ¼ 1,

...

... , 5, is presented by a shaded area in the figure.

A ¼fA \ B 1 g[fA \ B 2 g[fA \ B 3 g[fA \ B 4 g[fA \ B 5 g: (1.68)

Note than the events ( A\B i ) are mutually exclusive.

PfAg¼ X PfA \ B i g¼ X PfAB i g¼ X

5 5 5

PfAjB i gPfB i g: (1.69)

i¼ 1 i¼ 1 i¼ 1

Fig. 1.10 Illustration of total probability for i ¼ 1, ... ,5