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CHAPTER 1 Basic Probability
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Conditional Probability
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Let A and B be two events (Fig. 1-3) such that P(A) 0. Denote by P(B u A) the probability of B given that A has occurred. Since A is known to have occurred, it becomes the new sample space replacing the original S. From this we are led to the definition P(B u A) ; or P(A> B) P(A) (17) (18)
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P(A > B) ; P(A) P(B u A)
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Fig. 1-3
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In words, (18) says that the probability that both A and B occur is equal to the probability that A occurs times the probability that B occurs given that A has occurred. We call P(B u A) the conditional probability of B given A, i.e., the probability that B will occur given that A has occurred. It is easy to show that conditional probability satisfies the axioms on page 5.
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EXAMPLE 1.13 Find the probability that a single toss of a die will result in a number less than 4 if (a) no other information is given and (b) it is given that the toss resulted in an odd number.
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(a) Let B denote the event {less than 4}. Since B is the union of the events 1, 2, or 3 turning up, we see by Theorem 1-5 that P(B) P(1) P(2) P(3) 1 6 1 6 1 6 1 2
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assuming equal probabilities for the sample points. (b) Letting A be the event {odd number}, we see that P(A) P(B u A)
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3 6 1 2.
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Also P(A > B)
1 3.
Then
P(A > B) P(A)
1>3 1>2
Hence, the added knowledge that the toss results in an odd number raises the probability from 1 > 2 to 2 > 3.
Theorems on Conditional Probability
Theorem 1-9 For any three events A1, A2, A3, we have P(A1 > A2 > A3) P(A1) P(A2 u A1) P(A3 u A1> A2) (19)
In words, the probability that A1 and A2 and A3 all occur is equal to the probability that A1 occurs times the probability that A2 occurs given that A1 has occurred times the probability that A3 occurs given that both A1 and A2 have occurred. The result is easily generalized to n events. Theorem 1-10 If an event A must result in one of the mutually exclusive events A1, A2, c, An, then P(A) P(A1) P(A u A1) P(A2 ) P(A u A2) c P(An ) P(A u An ) (20)
Independent Events
If P(B u A) P(B), i.e., the probability of B occurring is not affected by the occurrence or non-occurrence of A, then we say that A and B are independent events. This is equivalent to P(A > B) P(A)P(B) (21)
as seen from (18). Conversely, if (21) holds, then A and B are independent.
CHAPTER 1 Basic Probability
We say that three events A1, A2, A3 are independent if they are pairwise independent: P(Aj > Ak ) and P(Aj)P(Ak ) P(A1 > A2 > A3) j2k where j, k 1, 2, 3 (22) (23)
P(A1)P(A2 )P(A3 )
Note that neither (22) nor (23) is by itself sufficient. Independence of more than three events is easily defined.
Bayes Theorem or Rule
Suppose that A1, A2, c, An are mutually exclusive events whose union is the sample space S, i.e., one of the events must occur. Then if A is any event, we have the following important theorem: Theorem 1-11 (Bayes Rule): P(Ak u A) P(Ak) P(A u Ak)
(24)
a P(Aj) P(A u Aj)
This enables us to find the probabilities of the various events A1, A2, c, An that can cause A to occur. For this reason Bayes theorem is often referred to as a theorem on the probability of causes.
Combinatorial Analysis
In many cases the number of sample points in a sample space is not very large, and so direct enumeration or counting of sample points needed to obtain probabilities is not difficult. However, problems arise where direct counting becomes a practical impossibility. In such cases use is made of combinatorial analysis, which could also be called a sophisticated way of counting.