Probability (2) basics of probability of an event

# Probability (2) basics of probability of an event - Review...

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Review be a measure of the possibility that a random phenomenon occurs, to find the regularity of these random phenomena . Probability Basics of probability theory Sample space Element (or sample point) Event Empty set Complement Intersection: Disjoint, mutually exclusive Union: exhaustive

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Review Venn diagram S A B A A’ S A B B A A S A B S S S A So, A and B are disjoint. S B A B AA S Exhaustive AND Mutually exclusive A1 A2 A3 A4 A B A, B, C, D and E are mutually exclusive. E D C A1 A2 A3 A4 S A A A A = 4 3 2 1 3 3 3
Probability of an event Since an event depends on unpredicted outcomes, it is random. Now we want to assign a value to quantify the possibility that an event occurs , where this value is determined by a function defined on the class of all events and the function is assumed to satisfy the following axioms . An axiom is a proposition regarded as self-evidently true without proof .

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Probability of an event Consider a random experiment whose sample space is S. Suppose that a function P(·) is defined on the class of all events and satisfies P(S) = 1; For any mutually exclusive events A 1 , A 2 , …. + + = ) ( ) ( ) ( 2 1 2 1 A P A P A A P For the function P(·) satisfying the above axioms, we call P(A) a probability that an event A occurs . 0 ≤ P(A) ≤ 1, for every event A of S;
Example } { } { } , { T H T H S A = = Consider the experiment of tossing a coin. Suppose that a head is as likely to appear as a tail. Then what is the probability that a head occurs? Denote the probability that a head and a tail occurs by P({ H }) and P({T}), respectively. Note that and events { H } and { T } are mutually exclusive.

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Example }) ({ }) ({ ) ( 1 T P H P S P + = = So by the first and the third axioms , we have . 2 1 }) ({ }) ({ = = T P H P Since a head is as likely to appear as a tail , P ({ H }) = P ({ T }). Thus, we have Consider the experiment of tossing a coin. Suppose that a head is as likely to appear as a tail. Then what is the probability that a head occurs?
Basics operations with probabilities . ) ( ) ( ) ( ) ( B A P B P A P B A P - + = If A and B are any two events, then

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S A B A B A B B A B B A A . ) ( ) ( ) ( ) ( B A P B P A P B A P - + =
Basics operations with probabilities ). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( C B A P A C P C B P B A P C P B P A P C B A P + - - - + + = More general, for three events A , B and C ,

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Probability (2) basics of probability of an event - Review...

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