Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 21

Answer
(a) From ( 1.76 ) the probability that all three messages are received correctly is:

PfDg¼p 1 3 : (1.100)

(b) From ( 1.13 ) the probability that each message is received either correctly or
partially correctly (mutually exclusive events) is equal to the sum of p 1 + p 2 .
Consequently, the probability that all three messages are received either
correctly or partially correctly is equal to ( p 1 + p 2 ) 3 . Note that this event is a
complement of event E . Therefore, from ( 1.18 ) we arrive at:

3

PfEg¼ 1 PfEg¼ 1 ðp 1 þ p 2 Þ : (1.101)

Exercise E.1.6 There are n 1 students in a group that usually obtain “excellent”
grades, n 2 students that usually obtain “good” grades, and n 3 students that usually
obtain “fair” grades. On a test, the students, that usually get “excellent” grades,
have equal probabilities of getting “excellent” and “good” grades. Similarly, those,
who usually get “good” grades, have equal probabilities of getting “excellent,”
“good,” or “fair” marks, and those, who usually get “fair” grades, have equal
probabilities of receiving “good” or “fair” grades.
A student is randomly selected. Find the probability of the event:

A ¼f student receives a ''good'' grade g: (1.102)

Answer We define the following events,

A 1 ¼f a student who usually gets ''excellent'' grades is selected g;
A 2 ¼f a student who usually gets ''good'' grades is selected g;
A 3 ¼f a student who usually gets ''fair'' grades is selected g: (1.103)

Knowing that the total number of students is,

n ¼ n 1 þ n 2 þ n 3 ; (1.104)

using ( 1.34 ) we find the probabilities of the events A i , i ¼ 1, ... ,3,

PfA 1 g¼n 1 =n:
PfA 2 g¼n 2 =n:
PfA 3 g¼n 3 =n:

(1.105)

The corresponding conditional probabilities are:

PfAjA 1 g¼ 1 = 2 ; PfAjA 2 g¼ 1 = 3 ; PfAjA 3 g¼ 1 = 2 : (1.106)