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 disjointness. 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?
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 disjointness. 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?