Page 46 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 29

(e) AB A [ B.
(f) AB) AC) ˆ;.
2.2 The second relation in Equations (2.10) expresses the union of two sets as the union
of two disjoint sets (i.e. A [ B ˆ A ‡ AB ). Express A [ B [ C in terms of the union
of disjoint sets where A, B, and C are arbitrary sets.
2.3 Verify DeMorgan’s laws, given by the last two equations of Equations (2.10).
2.4 Let S ˆf1, 2, ... ,10g, A ˆf1, 3, 5g, B ˆf1, 4, 6g, and C ˆf2, 5, 7g. Determine
elements of the following sets:
elements of the following sets:
(a) S [ C.
(b) A [ B.
(c) AC.
(d) A [ BC).
(e) ABC.
(f) AB.
(g) AB) [ BC) [ CA).
peat Problem 2.4
2.5 Repeat Problem 2.4 if if S ˆfx:0 x 10g, A ˆfx:1 x 5g, B ˆfx:1 x 6g,
and C ˆfx:2 x 7g.
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 P A) ˆ a, P B) ˆ b, and P C) ˆ c.
Determine the following probabilities in terms of a, b, and c:
(a) P AB).
(b) P A [ B).
(c) P A [ BjB).
(d) P A [ BjC).
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, and B:
(a) At least one of the components is good.
(b) One is good and one is defective.

TLFeBOOK