ssrs 2008 r2 barcode font The Axioms of Probability in Software

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The Axioms of Probability
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Suppose we have a sample space S. If S is discrete, all subsets correspond to events and conversely, but if S is nondiscrete, only special subsets (called measurable) correspond to events. To each event A in the class C of events, we associate a real number P(A). Then P is called a probability function, and P(A) the probability of the event A, if the following axioms are satisfied. Axiom 1 For every event A in the class C, P(A) Axiom 2 For the sure or certain event S in the class C, P(S) P(A1 < A2 < c ) P(A1 < A2) 1 c (2) Axiom 3 For any number of mutually exclusive events A1, A2, c, in the class C, P(A1) P(A2) (3) 0 (1)
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In particular, for two mutually exclusive events A1, A2, P(A1) P(A2) (4)
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Some Important Theorems on Probability
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From the above axioms we can now prove various theorems on probability that are important in further work. Theorem 1-1 If A1 ( A2, then P(A1) Theorem 1-2 For every event A, 0 P(A) P(\) 1, (5) P(A2) and P(A2 A1) P(A2) P(A1).
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i.e., a probability is between 0 and 1. Theorem 1-3 0 (6) i.e., the impossible event has probability zero.
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CHAPTER 1 Basic Probability
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Theorem 1-4 If Ar is the complement of A, then P(Ar) Theorem 1-5 If A 1 P(A) c (7) A1 < A2 < c < An, where A1, A2, . . . , An are mutually exclusive events, then P(A) In particular, if A P(A1) P(A2) c P(An) (8)
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S, the sample space, then P(A1) P(A < B) P(A2) P(An) 1 (9)
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Theorem 1-6 If A and B are any two events, then P(A) P(B) P(A > B) (10)
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More generally, if A1, A2, A3 are any three events, then P(A1 < A2 < A3) P(A1) P(A2) P(A3) P(A2 > A3) P(A3 > A1) (11) P(A1 > A2)
P(A1 > A2 > A3) Generalizations to n events can also be made. Theorem 1-7 For any events A and B, P(A) P(A > B) P(A > Br)
(12)
Theorem 1-8 If an event A must result in the occurrence of one of the mutually exclusive events A1, A2, . . . , An, then P(A) P(A > A1) P(A > A2) c P(A > An) (13)
Assignment of Probabilities
If a sample space S consists of a finite number of outcomes a1, a2, c , an, then by Theorem 1-5, P(A1) P(A2) c P(An) 1 (14) where A1, A2, c , An are elementary events given by Ai {ai}. It follows that we can arbitrarily choose any nonnegative numbers for the probabilities of these simple events as long as (14) is satisfied. In particular, if we assume equal probabilities for all simple events, then 1 n , k 1, 2, c, n and if A is any event made up of h such simple events, we have P(Ak) P(A) (15)
h (16) n This is equivalent to the classical approach to probability given on page 5. We could of course use other procedures for assigning probabilities, such as the frequency approach of page 5. Assigning probabilities provides a mathematical model, the success of which must be tested by experiment in much the same manner that theories in physics or other sciences must be tested by experiment.
EXAMPLE 1.12
A single die is tossed once. Find the probability of a 2 or 5 turning up. {1, 2, 3, 4, 5, 6}. If we assign equal probabilities to the sample points, i.e., if we assume that
The sample space is S the die is fair, then
P(1)
P(2)
P(6)
The event that either 2 or 5 turns up is indicated by 2 < 5. Therefore, P(2 < 5) P(2) P(5) 1 6 1 6 1 3
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