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1. Nations of Probability 9
Example 1.3.3 In a college campus, suppose that 2600 are women out of
4000 undergraduate students, while 800 are men among 2000 undergraduates
who are under the age 25. From this population of undergraduate students if
one student is selected at random, what is the probability that the student will
be either a man or be under the age 25? Define two events as follows
A: the selected undergradute student is male
B: the selected undergradute student is under the age 25
and observe that P(A) = 1400/4000, P(B) = 2000/4000, P(A ∩ B) = 800/4000.
Now, apply the Theorem 1.3.3, part (iv), to write P(A ∪ B) = P(A) + P(B)
P(A ∩ B) = (1400 + 2000 800)/4000 = 13/20. !
Having a sample space S and appropriate events from a Borel
sigma-field ß of subsets of S, and a probability scheme satisfying
(1.3.1), one can evaluate the probability of the legitimate events
only. The members of ß are the only legitimate events.
1.4 The Conditional Probability and Independent
Events
Let us reconsider the Example 1.3.2. Suppose that the two fair dice, one red
and the other yellow, are tossed in another room. After the toss, the experi-
menter comes out to announce that the event D has been observed. Recall that
P(E) was 1/9 to begin with, but we know now that D has happened, and so the
probability of the event E should be appropriately updated. Now then, how
should one revise the probability of the event E, given the additional informa-
tion?
The basic idea is simple: when we are told that the event D has been ob-
served, then D should take over the role of the sample space while the origi-
nal sample space S should be irrelevant at this point. In order to evaluate the
probability of the event E in this situation, one should simply focus on the
portion of E which is inside the set D. This is the fundamental idea behind the
concept of conditioning.
Definition 1.4.1 Let S and ß be respectively the sample space and the
Borel sigma-field. Suppose that A, B are two arbitrary events. The condi-
tional probability of the event A given the other event B, denoted by P(A | B),
is defined as
