Discrete Mathematics for Computer Science

Independent Events and Conditional Probability 509

Solution. The sample space is Ž = {l, 2, 3, 4, 5, 61. P(A) = P(B) =1/2, but P(C) =

2/3. Events A, B, and C are independent, because

1

P(A n Bn C) = P({1})=- 6

and

P(A). P(B). P(C)= (1)(1)(2)

Events A and C are not independent, because

1

P(A n C) = P({1, 2,31) = -

2

but

P(A). P(C)= ()(2)

2 33

Events B and C are independent, because

1

P(B n C) = P({1,4}) 3

and

P(B). P(C) = (=)(2)

2 33

We now list several important points to remember about independence of events. (The

reader is asked to prove some of these in Section 8.6.)

Independence

"* Disjointness and independence are different concepts. In fact, disjoint events that

have positive probabilities are never independent.

"* Events in the individual sample spaces of unrelated experiments give rise to inde-

pendent events in the cross product of the individual sample spaces.

"* Independence is a property of a collection of two or more events, not a property of

just one event.

"* A collection of pairwise independent events does not always constitute an indepen-

dent set of events.

"* In an independent set of events, not all the pairs need be independent. (See Exam-

ple 2).

8.5.2 Introduction to Conditional Probability

In Example 2 of Section 8.5.1, it was found that for the sample space Q2 = {1, 2, 3, 4, 5, 6}

the events B = {1, 4, 5} and C = {1, 2, 3, 41 were, mathematically speaking, independent

even though they arise from the same experiment. Events B and C have nothing to do with