Discrete Mathematics for Computer Science

Independent Events and Conditional Probability 515

Send 0 with 0.8

probability 0.4 o Receive^0

0.2 0.1

Send 1 with -09 l Receive 1

probability 0.6

Figure 8.5 Communication channel reliability.

Solution. Let B denote the event that a 1 was sent, and let A denote the event that a I

was received. The probability we seek is P(B IA), which according to Bayes' Rule is

P(BI A) - P(A I B) .P(B) _ (0.9)(0.6)

P(A) P(A)

What is P(A), the probability that a 1 was received? We are not given this information
directly, but we can compute it using the Theorem of Total Probability.
To apply the Theorem of Total Probability, we should check that we know what sample
space 02 we are using. We can take Q2 to be all 2-tuples of 0's and 1's where the first element
gives the transmitted bit and the second element gives the received bit. Note that Q2 is not
a cross product sample space, because the probability of receiving a 1, say, depends on the
bit that is transmitted.
Sample space Q2 can be regarded as the disjoint union of events with positive proba-
bilities as follows: Let E 1 be the event that a^1 was sent, and let E 2 be the event that a^0
was sent. Hence, 02 = El U E 2 .By the Theorem of Total Probability, the probability that
a 1 was received is

P(A) = P(A I El) " P(Et) + P(A I E 2 ). P(E 2 )

= (0.9)(0.6) + (0.2)(0.4)
= 0.62
Now, we can complete the computation of P (B I A), the event that a 1 was sent given
that a 1 was received:

(0.9)(0.6) (0.9)(0.6)

P(B I A) - P-_ P(A) 0.62 0.87 U

The communication channel reliability situation of Example^6 can be conceptualized
as a two-stage process. First, a bit is selected for transmission. The bit is either transmitted
correctly or incorrectly. This is not the same as flipping two coins, however, where the
result of one flip has no influence on the result of the other. Here, the probability that a
bit is transmitted without error depends on whether the bit is a^0 or a 1. As suggested in
Example 6, we can choose Q2 = (0, 0), (0, 1), (1,0), (1, 1)} where the first component of
each ordered pair gives the bit transmitted and the second component gives the bit received.
What probability density should we assign to this Q2? The next example shows that only
one choice is consistent with the data in Figure 8.5.

Example 7. (Communication Channel Reliability Continued) Suppose that the
communication channel reliability situation is modeled by the sample space^02 just sug-
gested. Determine a probability density on 02 that is consistent with the given data.