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Chapter 4 Probability: Probabilities of Compound Events 4.1THE ADDITION RULE 4.1.1 The General Addition Rule 4.1.2The Special Addition Rule for Mutually Exclusive Events 4.2 Conditional Probabilities 4.3 The Multiplication Rule 4.4 Independent Events and the Special Multiplication Rule
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4.1THE ADDITION RULE
4.1.1 The General Addition Rule
4.1.2The Special Addition Rule for Mutually Exclusive Events
4.2 Conditional Probabilities
4.3 The Multiplication Rule
4.4 Independent Events and the Special Multiplication Rule
4.4.1Independence of Two Events
4.4.2Independence of More Than Two Events and the Special Multiplication Rule
4.5 Bayes’ Theorem
4.5.1 The Total Probability
4.5.2 Bayes’ Theorem
The general addition rule for two events, A and B, in the sample space S:
P(AB) = P(A) + P(B) – P(AB)
classical, popular, rock, folk and jazz. The respective probabilities that a customer buying a record will choose from each section are 0.3, 0.4, 0.2, 0.05 and 0.05. Find the probability that a person (a) will choose a record from the classical or the folk or the jazz sections, (b) will not choose a record from the rock or folk or classical sections.
If A1, A2, …, Ak are mutually exclusive, then
P(A1A2…Ak) = P(A1) + P(A2) +… + P(Ak).
If A and B are two events and P(A) 0 and P(B) 0, then the probability of A, given that B has already occurred is written P(A|B) and
The general multiplication rule for events A and B in the sample space S:
P(AB) = P(A) P(B|A)
P(AB) = P(B) P(A|B)
P(ABC) = P(A) P(B|A) P(C|AB)
If the occurrence or non-occurrence of an event A does not influence in any way the probability of an event B, then event B is independent of event A and P(B|A) = P(B).
Two events A and B are independent iff P(AB) = P(A)P(B)
If k events A1, A2,…., Ak are independent, then
P(A1 A2…. Ak) = P(A1)P(A2)…P(Ak)
. Find also
Let the sample space S be partitioned into mutually exclusive events Ej’s (j = 1,2,…,k) and let A be an event in S. Then the probability of Er conditional on A is
P(Er |A) = for r =1,2,…,k
(I) What is the probability that a white ball is chosen?
(ii) Suppose a white ball is chosen, find the probability that this white ball comes from the 1st box.
(i) He was late for his visit. Find the probability that he had travelled by MTR.
(ii) He was not late for his visit. Find the probability that he had travelled by Bus.