Fundamentals of Probability and Statistics for Engineers

(e)

(f)

2.2 The second relation in Equations (2.10) expresses the union of two sets as the union
of two disjoint sets (i.e. ). Express in terms of the union
of disjoint sets where A, B, and C are arbitrary sets.

2.3 Verify D eM organ’s laws, given by the last two equations of Equations (2.10).

2.4 Let D etermine
elements of the following sets:
(a)
(b)
(c)
(d)
(e)
(f)
(g)

2.5 R epeat Problem 2.4 if
and

2.6 Draw Venn diagrams of events A and B representing the following situations:
(a) A and B are arbitrary.
(b) If A occurs, B must occur.
(c) If A occurs, B cannot occur.
(d) A and B are independent.

2.7 Let A, B, and C be arbitrary events. Find expressions for the events that of A, B, C:
(a) None occurs.
(b) Only A occurs.
(c) Only one occurs.
(d) At least one occurs.
(e) A occurs and either B or C occurs but not both.
(f) B and C occur, but A does not occur.
(g) Two or more occur.
(h) At most two occur.
(i) All three occur.

2.8 Events A, B, and C are independent, with
Determine the fo llowing probabilities in terms of a, b, and c:
(a)
(b)
(c)
(d)

2.9 An engineering system has two components. Let us define the following events:

A : first component is good;A: first component is defective.

B : second component is good;B: second component is defective:

Describe the following events in terms of A,A, B, andB:

(a) At least one of the components is good.

(b) One is good and one is defective.

Basic Probability Concepts 29

ABA[B.

AB) AC)ˆ;.

A[BˆA‡AB A[B[C

Sˆf1, 2,...,10g,Aˆf1, 3, 5g,Bˆf1, 4, 6g, and Cˆf2, 5, 7g.

elements of the following sets:

S[C.

A[B.

AC.

A[ BC).

ABC.

AB.

AB)[ BC)[ CA).

peat Problem 2.4 ifSˆfx: 0 x 10 g,Aˆfx: 1 x 5 g,Bˆfx: 1 x 6 g,

Cˆfx: 2 x 7 g.

P A)ˆa,P B)ˆb, andP C)ˆc.

P AB).

P A[B).

P A[BjB).

P A[BjC).