
Gaining partial information relevant to the
experiment’s outcome may cause us to revise
the probability of the events.
2

Components are assembled in a plant that uses
two assembly lines A and A. Line A uses older
equipment that is slower and less reliable.
Suppose that on a given day A has assembled 8
components, of which 2 are defective ( B ) and six
are nondefective ( B ). Line A has assembled 10
components, of which 1 is defective.
One of the 18 components is chosen at random.
The probability that the item came from line
A changes if we have the prior information that the
selected item is defective.

3



In effect, knowing that event B occurs
restricts the sample space to those outcomes
that are in B.
The outcomes that are of interest to event A
are those in both A and B.
Since the probabilities for simple events in B
sum to P(B), we need to re-normalize
probabilities by dividing by P(B).
4
 For
two events A and B with
P(B)>0, the conditional
probability of A given that
B has occurred is
P A  B
P( A B) 
P B
5

Suppose that of all individuals buying a certain
digital camera, 60% include an optional memory
card, 40% include an extra battery, and 30%
include both. Consider randomly selecting a
buyer and let A=(memory card purchased) and
B=(battery purchased). Let P(A)=0.6, P(B)=0.4
and P(both purchased)=0.3.
We compute
conditional probabilities for A given B and for B
given A. Notice that they are different from the
unconditional probabilities.
6

P A  B  P A B PB
7

Let A1 , A2 , , Ak be mutually exclusive events
whose union is the sample space (exhaustive).
Then for any other event B,
P  B    i1 P  B Ai P  Ai 
k
 P  B A1  P  A1  
 P  B Ak  P  Ak 
8



An individual has three different email
accounts. Of her messages, 70% come into
account 1, 20% into account 2, and 10% into
account 3.
Of the messages into account 1, only 1% are
spam, whereas the corresponding percentages
are 2% and 5% for accounts 2 and 3.
What is the probability that a randomly
selected message is spam?
9

Let A1 , A2 , , Ak be mutually exclusive and
exhaustive events with prior probabilities,
P  Ai  , i  1, , k . Then for any other event B
for which P( B)  0 , the posterior
probabilities are
P  Aj B  
P  Aj  B 
P B


P  Aj  P B Aj


P
A
P
B
A



i
i
i 1
k
10