b Which outcomes are in the event that exactly three of the selected mortgages

B which outcomes are in the event that exactly three

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(b) Which outcomes are in the event that exactly three of the selected mortgages are fixed rate? (c) Which outcomes are in the event that all four mortgages are of the same type? (d) Which outcomes are in the event that at most one of the four is a variable-rate mortgage? (e) What is the union of the events in parts (c) and (d), and what is the intersection of these two events? (f) What are the union and intersection of the two events in parts (b) and (c)? 31
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Chapter 2. Probability STAT 155 2.2 Axioms, Interpretations, and Properties of Probability Given an experiment and a sample space S , the objective of probability is to assign to each event A a number P ( A ) , called the probability of the event A , which will give a precise mea- sure of the chance that A will occur. Law of Large Numbers – the relative frequency of the number of times that an outcome occurs when an experiment is repli- cated over and over again approaches the theoretical proba- bility of the outcome. The probability should satisfy the following three axioms (ba- sic properties) : Axiom 1 For any event A , P ( A ) 0 . Axiom 2 P ( S ) = 1 . Axiom 3 If A 1 , A 2 , · · · , is an infinite collection of disjoint events, then P ( A 1 A 2 A 3 ∪ · · · ) = X i =1 P ( A i ) . 32
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Chapter 2. Probability STAT 155 Properties of Probability P (Ø) = 0 A 1 , · · · , A k are disjoint events, then P ( A 1 A 2 A 3 · · · A k ) = k X i =1 P ( A i ) P ( A ) + P ( A 0 ) = 1 0 P ( A ) 1 33
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Chapter 2. Probability STAT 155 For any two events A and B , P ( A B ) = P ( A ) + P ( B ) - P ( A B ) If events A and B are disjoint events, then P ( A B ) = P ( A ) + P ( B ) . For any three events A, B, and C , P ( A B C ) = P ( A ) + P ( B ) + P ( C ) - P ( A B ) - P ( A C ) - P ( B C ) + P ( A B C ) 34
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Chapter 2. Probability STAT 155 Example : Suppose that A and B are two events. P ( A ) = 0 . 8 , P ( B ) = 0 . 7 . (a) Is it possible that P ( A B ) = 0 . 1 ? (b) What is the smallest possible value for P ( A B ) ? (c) Is it possible that P ( A B ) = 0 . 77 ? Exercise 2.12 Consider randomly selecting a student at a certain univer- sity, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard. Suppose that P ( A ) = . 5 , P ( B ) = . 4 , and P ( A B ) = . 25 . (a) Compute the probability that the selected individual has at least one of the two types of cards (i.e., the probability of the event A B ). (b) What is the probability that the selected individual has neither type of card? (c) Describe, in terms of A and B , the event that the selected student has a Visa card but not a MasterCard, and then calculate the probability of this event. 35
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Chapter 2. Probability STAT 155 When the sample space S is either finite or “countably infinite”, the probability of any event A is computed by adding together the individual probabilities for each outcome in A . If A = { E 1 , E 2 , · · · , E k } , then P ( A ) = k i =1 P ( E i ) .
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