B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

8.1 Concept of Probability 405
Figure 8.4
The event of
interest B and the
partition of S by
{A;}.
Total Probability Theorem: Let n disjoint events A1, ..• , An form a partition of the
sample space S such that
n
LJA; = S and A; nAj = 0, if i fcj
i=l
Then the probability of an event B can be written as
n
P(B) = L P(BIA;)P(A;)
i=l
Proof The proof of this theorem is quite simple based on Fig. 8.4. Since {A;} form a partition
of S, then
B=BnS = Bn(A1 UA2 U·•·UA n )
= (A 1 B) U (A2B) U · · · U (AnB)
Because {A;} are disjoint, so are {A;B}. Thus,
n
P(B) = L P(A;B) = L P(BIA;)P(A;)
i=l
This theorem can simplify the analysis of the more complex event of interest B by identifying
all different causes A; for B. By quantifying the effect of A; on B through P(BIA;), the
theorem allows us to "divide-and-conquer" a complex problem ( of event B).
11
i=l
Example 8.10 The decoding of a data packet may be in error because of N distinct error patterns
E1, E2, ... ,EN it encounters. These error patterns are mutually exclusive, each with probability
P(E;) = p;. When the error pattern E; occurs, the data packet would be incorrectly
decoded with probability q;. Find the probability that the data packet is incorrectly decoded.
We apply total probability theorem to tackle this problem. First, define B as the event that
the data packet is incorrectly decoded. Based on the problem, we know that
P(BjE;) = q; and P(E;) = p;