Page 90 - Fundamentals of Communications Systems

3.4 Chapter Three

EXAMPLE 3.5
Example 3.1(cont.). If the die is rolled and you are told the outcome is even, A 2 , how
does that change the probability of event A 3 ?
1
P (A 2 ∩ A 3 ) 6 1
P [A 3 |A 2 ] = = =
P (A 2 ) 1 3
2
Since P [A 3 |A 2 ] = P [A 3 ], event A 3 is not independent of event A 2 .

Definition 3.5 A collectively exhaustive set of events is one for which

A 1 ∪ A 2 ∪ .... ∪ A N =
Theorem 3.2 Total Probability For N mutually exclusive, collectively exhaustive
events (A 1 , A 2 , ..... A N ) and B ∈ , then
N

P [B] = P [B|A i ]P [A i ]
i=1

Proof: The probability of event B can be written as
N N

P [B] = P [B ∩ ] = P B ∩ A i = P B ∩ A i
i=1 i=1
The events B ∩ A i are mutually exclusive, so

N N N

P [B] = P B ∩ A i = P [B ∩ A i ] = P [B|A i ]P [A i ]
i=1 i=1 i=1
EXAMPLE 3.6
Example 3.1(cont.). Note A 2 and A 4 are a set of mutually exclusive collectively exhaust-
ive events with
2
P [A 3 ∩ A 4 ] P [{1, 3}] 6 2
P [A 3 |A 4 ] = = = =
P [A 4 ] P [A 4 ] 1 3
2
This produces

1 1 2 1 3 1
P [A 3 ] = P [A 3 |A 2 ]P (A 2 ) + P [A 3 |A 4 ]P (A 4 ) = + = =
3 2 3 2 6 2

Theorem 3.3 (Bayes) For N mutually exclusive, collectively exhaustive events
{A 1 , A 2 , ... A N } and B ∈ , then the conditional probability of the event A j given that