Conditional Probability
If two events E 1 and E 2 are related in some manner such that the probability of
occurrence of event E 1 depends upon whether E 2 has or has not occurred, then this is
called the conditional probability of E 1 given E 2 and it is denoted using the notation
P(E 1 | E 2 ). Here the vertical line is read as “given”and events to the right of the
vertical line are treated as events which have occurred. The conditional probability
of E 1 given E 2 is
P(E 1 |E 2 ) = P(E^1 ∩E^2 )
P(E 2 )
, P (E 2 )= 0 (11 .23)
The conditional probability of E 2 given E 1 is
P(E 2 |E 1 ) =
P(E 1 ∩E 2 )
P(E 1 ) , P (E^1 )= 0 (11 .24)
The equations (11.23) and (11.24) imply that the probability of both events E 1 and
E 2 occurring is given by
P(E 1 ∩E 2 ) = P(E 1 )P(E 2 |E 1 ) = P(E 2 )P(E 1 |E 2 ), P (E 1 )= 0, P (E 2 )= 0 (11.25)
If the events E 1 and E 2 are independent events, then
P(E 1 ∩E 2 ) = P(E 1 )P(E 2 ) (11 .26)
and consequently
P(E 1 |E 2 ) = P(E 1 ), and P(E 2 |E 1 ) = P(E 2 )
This condition occurs whenever the probability of E 1 does not depend upon the event
E 2 and similarly, the probability of event E 2 does not depend upon E 1.
Two events E 1 and E 2 are said to be independent events if and only if the
probability of occurrence of E 1 and E 2 is given by P(E 1 ∩E 2 ) = P(E 1 )P(E 2 ). That is,
the probability of both E 1 and E 2 occurring is the product of the probabilities of
occurrence of each event. Two events that are not independent are called dependent
events.
In general, if E 1 , E 2 ,... , E mare all independent events, then
P(E 1 ∩E 2 ∩···∩ Em) = P(E 1 )P(E 2 )···P(Em) (11 .27)
This is sometime written in the form
P( m∩
k=1
Ek) =
m
Π
k=1
P(Ek) (11 .28)
and is known as the multiplication principle for independent events.
