Page 110 - Fundamentals of Communications Systems

3.24 Chapter Three

(a) For design purposes it is typical to model a patient’s requested service as a
random experiment. Define a sample space for the requested services.

A market survey has shown that the probability that a patient will request
an EKG is 0.4 and the probability that a patient will request 911 is 0.001. The
probability that a digital service will be requested is 0.1.

(b) Compute the probability of requesting a doctor’s appointment.
(c) Compute the probability of requesting an audio transmission from a
stethoscope.

Problem 3.2. This problem exercises the idea of conditional probability. In the
Problem 3.1 if you know the requested service is a digital service what is the
probability that a 911 request is initiated?

Problem 3.3. A simple problem to exercise the axioms of probability. Two events
A and B are defined by the same random experiment. P (A) = 0.5, P (B) = 0.4,
and P (A∩ B) = 0.2

(a) Compute P (A∪ B).
(b) Are events A and B independent?

Problem 3.4. Your roommate challenges you to a game of chance. He proposes
the following game. A coin is flipped two times, if heads comes up twice she/he
gets a dollar from you and if tails comes up twice you get a dollar from him/her.
You know your roommate is a schemer so you know that there is some chance
that this game is rigged. To this end you assign the following mathematical
framework.

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■ P (F ) = probability that the game is fair =
6
■ P (H|F ) = probability that a head results if the game is fair = 0.5
■ P (H|UF ) = probability that a head results if the game is unfair = 0.75

Assume conditioned on the fairness of the game that each flip of the coin is
independent.

(a) What is the probability that the game is unfair, P (UF )?
(b) What is the probability that two heads appear given the game is unfair?
(c) What is the probability that two heads appear?
(d) If two heads appear on the first trial of the game, what is the probability
that you are playing an unfair game?

Problem 3.5. Certain digital communication schemes use redundancy in the form
of an error control code to improve the reliability of communication. The com-
pact disc recording media is one such example. Assume a code can correct two