Page 111 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.25

or fewer errors in a block of N coded bits. If each bit detection is independent
with a probability of error of P (E) = 10 −3

1. Plot the probability of correct block detection as a function of N (Note, it is
most useful to have log scaling on the y-axis).
2. How small must N be so that the probability the block is detected incorrectly
−6
is less than 10 ?
Problem 3.6. The probability of accessing a campus computer system is charac-
terized on a weekday with the following events

1. A ={dialing is between 7:00PM–1:00AM}
2. B ={dialing is between 1:00AM–7:00PM}
3. C ={a connection is made}
4. D ={a connection failed}

The system is characterized with
P (Connecting given dialing is between 7:00PM–1:00AM) = P (C|A) = 0.1
P (Connecting given dialing is between 1:00AM–7:00PM) = P (C|B) = 0.75

(a) If a person randomly dials with uniform probability during the day what is
P (A)?
(b) If a person randomly dials with uniform probability during the day what
is P (C)?
(c) If a person randomly dials with uniform probability during the day what is
the P (A and C)?
(d) Compute P (A|C).
Problem 3.7. In a particular magnetic disk drive bits are written and read indi-
−8
vidually. The probability of a bit error is P (E) = 10 . Bit errors are indepen-
dent from bit to bit. In a computer application the bits are grouped into 16-bit
words. What is the probability that an application word will be in error (at least
one of the 16 bits is in error) when the bits are read from this magnetic disk
drive.
Problem 3.8. A simple yet practical problem to exercise the idea of Bayes rule
and total probability. Air traffic control radars (ATCR) are used to direct traffic
in and around airports. If a plane is detected where it is not suppose to be then
the controllers must initiate some evasive actions. Consequently, for the users
of a radar system the two most important parameters of a radar system are

■ P (TA|DT ) = probability of target being absent when a target is detected.
■ P (TP|ND) = probability of target being present when no detection is
obtained.