Discrete Mathematics for Computer Science

CHAPTER 8 Discrete Probability

By Corollaries 1 and 2 of Theorem 3 in Section 8.3.4, pairs of events of the form
E and E for i A j in the sample space 0 = Q, X 02 x ... x Qn for some n E N are
examples of independent pairs of events in a cross product sample space. It is also possible
for events that do not have this form to be independent.

Example 1. Consider one roll of a fair die with Q2 = {1, 2, 3, 4, 5, 61. Let A be the event
that an even number is rolled. Let B be the event that the number rolled is at most two. Let
C be the event that the number rolled is a prime. Which pairs of events are independent?

Solution. A = t2, 4, 61, B = {1, 2), and C = {2, 3, 51. Events A and B are independent:

1

P(A n B) = P({2j) = - 6

and

P(A). P(B) = (- )•) -^6

On the other hand, A and C are not independent:

1

P(A n C) = P({2}) = - 6

but

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

2 2 4

It is important not to confuse the concept of independence with the concept of disjoint-

ness. In fact, disjoint events that have positive probabilities are never independent. As we

have seen, physically unrelated experiments give rise to a cross product sample space in

which nonempty events Ei on the various sample spaces Qi can be regarded as events E*

in a common sample space

"1 X 02 X ... X Qn

Furthermore, as we pointed out following Definition 1, these events are, mathematically

speaking, independent. However, they are not disjoint:

E* n Ej* = 21 x... x Ei x... x Ej x... x Qn

The next definition extends the concept of independence from pairs to sets of more

than two events belonging to the same sample space.

Definition 2. Let A 1 , A 2 ... Ak be subsets of the same sample space. The set of events

{A 1 , A 2 ... , Akd is called an independent set of events provided that

P(A 1 l n A 2 n. .. fn Ak) = H P(Ai)

The definition will be used in Section 8.9.2.

Example 2. A fair die is rolled one time. Let A denote the event (1, 2, 31, B the event {1,

4, 5), and C the event {1, 2, 3, 4}. Are A, B, and C an independent set of events? Are A

and C independent? Are B and C independent?