Page 33 - Probability and Statistical Inference

10 1. Nations of Probability

In the same vein, we will write P(B | A) = P(A ∩ B) | P(A) provided that
P(A) > 0.
Definition 1.4.2 Two arbitrary events A and B are defined independent if
and only if P(A | B), that is having the additional knowledge that B has been
observed has not affected the probability of A, provided that P(B) > 0. Two
arbitrary events A and B are then defined dependent if and only if P(A | B) ≠
P(A), in other words knowing that B has been observed has affected the prob-
ability of A, provided that P(B) > 0.
In case the two events A and B are independent, intuitively it means that
the occurrence of the event A (or B) does not affect or influence the probabil-
ity of the occurrence of the other event B (or A). In other words, the occur-
rence of the event A (or B) yields no reason to alter the likelihood of the other
event B (or A).
When the two events A, B are dependent, sometimes we say that B is favor-
able to A if and only if P(A | B) > P(A) provided that P(B) > 0. Also, when the
two events A, B are dependent, sometimes we say that B is unfavorable to A if
and only if P(A | B) < P(A) provided that P(B) > 0.
Example 1.4.1 (Example 1.3.2 Continued) Recall that P(D ∩ E) = P(53) =
1/36 and P(D) = 5/36, so that we have P(E | D) = P(D∩ E) | P(D) = 1/5. But,
P(E) = 1/9 which is different from P(E | D). In other words, we conclude that D
and E are two dependent events. Since P(E | D) > P(E), we may add that the
event D is favorable to the event E. !
The proof of the following theorem is left as the Exercise 1.4.1.
Theorem 1.4.1 The two events B and A are independent if and only if A
and B are independent. Also, the two events A and B are independent if and
only if P(A ∩ B) = P(A) P(B).
We now state and prove another interesting result.
Theorem 1.4.2 Suppose that A and B are two events. Then, the following
statements are equivalent:
(i) The events A and B are independent;
(ii) The events A and B are independent;
c
(iii) The events A and B are independent;
c
(iv) The events A and B are independent.
c
c
Proof It will suffice to show that part (i) ⇒ part (ii) ⇒ part (iii) ⇒ part (iv)
⇒ part (i).
(i) ⇒ (ii) : Assume that A and B are independent events. That is, in
view of the Theorem 1.4.1, we have P(A ∩ B) = P(A)P(B). Again in view
of the Theorem 1.4.1, we need to show that P(A ∩B) = P(A )P(B). Now,
c
c