Page 89 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.3

EXAMPLE 3.3

P [A∪ B] = P [A] + P [B] − P [A∩ B]

P [A∪ B ∪ C] = P [A] + P [B] + P [C] − P [A∩ B]

− P [B ∩ C] − P [A∩ C] + P [A∩ B ∩ C] (3.7)

EXAMPLE 3.4
Example 3.1(cont.).

P [A 2 ∪ A 3 ] = P [A 2 ] + P [A 3 ] − P [A 2 ∩ A 3 ] (3.8)
1
A 2 ∩ A 3 = 2 P [A 2 ∩ A 3 ] = (3.9)
6
1 1 1 5
P [A 2 ∪ A 3 ] = P [1, 2, 3, 4, 6] = + − = (3.10)
2 2 6 6

Definition 3.3 (Conditional Probability) Let ( , F, P ) be a probability space with sets
A, B ∈ F and P [B] = 0. The conditional or a posteriori probability of event A given an
event B, denoted P [A|B], is defined as

P [A∩ B]
P [A|B] =
P [B]
P [A|B] is interpreted as event A’s probability after the experiment has been
performed and event B is observed.

Definition 3.4 (Independence) Two events A, B ∈ F are independent if and only if

P [A∩ B] = P [A]P [B]

Independence is equivalent to

P [A|B] = P [A] (3.11)

For independent events A and B, the a posteriori or conditional probability
P [A|B] is equal to the a prioir probability P [A]. Consequently, if A and B are
independent, then observing B reveals nothing about the relative probability
of the occurrence of event A.