STAT 400 Lecture AL1 Examples for 1.1 Spring 2015 Dalpiaz Complement of A A' ( not A , A, Ac ) contains all elements that are not in A Intersection of A and B A ∩ B ( A and B , A B ) contains all elements that are in A and in B Union of A and B A ∪ B ( A or B ) contains all elements that are either in A or in B or both Axiom 1 Let A be any event defined over S. Then P ( A ) ≥ 0. Axiom 2 P ( S ) = 1. Axiom 3 If A 1 , A 2 , A 3 , … are events and A i ∩ A j = ∅ for each i ≠ j, then P ( A 1 ∪ A 2 ∪ … ∪ A k ) = P ( A 1 ) + P ( A 2 ) + … + P ( A k ) for each positive integer k, and P ( A 1 ∪ A 2 ∪ A 3 ∪ … ) = P ( A 1 ) + P ( A 2 ) + P ( A 3 ) + … for an infinite, but countable, number of events.Theorem 1. P ( A' ) = 1 – P ( A ). Theorem 2. P ( ∅ ) = 0. Theorem 3. If A ⊂ B, then P ( A ) ≤ P ( B ). Theorem 4. For any event A, P ( A ) ≤ 1. ! For any event A, 0 ≤ P(A) ≤ 1 ! P(S) = 1, where S is the sample space. ! Theorem 5. If A and B are any two events, then P(A ∪ B) = P(A) + P(B) – P(A ∩ B). P(A or B) = P(A) + P(B) – P(A and B). Theorem 6. P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C ) – P ( A ∩ B ) – P ( A ∩ C ) – P ( B ∩ C ) + P ( A ∩ B ∩ C ) P ( A ∪ B ∪ C ∪ D ) = P ( A ) + P ( B ) + P ( C ) + P ( D ) – P ( A ∩ B ) – P ( A ∩ C ) – P ( A ∩ D ) – P ( B ∩ C ) – P ( B ∩ D ) – P ( C ∩ D ) + P ( A ∩ B ∩ C ) + P ( A ∩ B ∩ D ) + P ( A ∩ C ∩ D ) + P ( B ∩ C ∩ D ) – P ( A ∩ B ∩ C ∩ D ) • • •1. Suppose a 6-sided die is rolled. The sample space, S , is { 1, 2, 3, 4, 5, 6 }. Consider the following events: A = { the outcome is even }, B = { the outcome is greater than 3 }, a) List outcomes in A, B, A', A ∩ B, A ∪ B. b) Find the probabilities P( A ), P( B ), P( A' ), P( A ∩ B ), P( A ∪ B ) if the die is balanced (fair). c) Suppose the die is loaded so that the probability of an outcome is proportional to the outcome, i.e. P( 1 ) = p, P( 2 ) = 2 p, P( 3 ) = 3 p, P( 4 ) = 4 p, P( 5 ) = 5 p, P( 6 ) = 6 p. i) Find the value of p that would make this a valid probability model. ii) Find the probabilities P( A ), P( B ), P( A' ), P( A ∩ B ), P( A ∪ B ).2. Consider a “thick” coin with three possible outcomes of a toss ( Heads, Tails, and Edge ) for which Heads and Tails are equally likely, but Heads is five times as likely than Edge. What is the probability of Heads ? 3. The probability that a randomly selected student at Anytown College owns a bicycle is 0.55, the probability that a student owns a car is 0.30, and the probability that a student owns both is 0.10. a) What is the probability that a student selected at random does not own a bicycle? b) What is the probability that a student selected at random owns either a car or a bicycle, or both? c) What is the probability that a student selected at random has neither a car nor a bicycle? C C ' B B'4. During the first week of the semester, 80% of customers at a local convenience store bought either beer or potato chips (or both). 60% bought potato chips. 30% of the customers bought both beer and potato chips. What proportion of customers bought beer? 5. Suppose P ( A ) = 0.22, P ( B ) = 0.25, P ( C ) = 0.28, P ( A ∩ B ) = 0.11, P ( A ∩ C ) = 0.05, P ( B ∩ C ) = 0.07, P ( A ∩ B ∩ C ) = 0.01. Find the following: a) P ( A ∪ B ) b) P ( A' ∩ B' ) c) P ( A ∪ B ∪ C ) d) P ( A' ∩ B' ∩ C' ) e) P ( A' ∩ B' ∩ C ) f) P ( ( A' ∩ B' ) ∪ C ) g) P ( ( A ∪ B ) ∩ C ) h) P ( ( B ∩ C' ) ∪ A' )6. Let a > 2. Suppose S = { 0, 1, 2, 3, … } and P( 0 ) = c, P( k ) = ka 1, k = 1, 2, 3, … . a) Find the value of c ( c will depend on a ) that makes this is a valid probability distribution. b) Find the probability of an odd outcome. 7. Suppose S = { 0, 1, 2, 3, … } and P( 0 ) = p, P( k ) = ! 21 kk⋅, k = 1, 2, 3, … . Find the value of p that would make this a valid probability
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