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Statistics - Practice Problems Mutually exclusive or disjoint events, dependent and independent events (Nonempty events A and B) Mutually exclusive or disjoint events: S P( A and B ) 0 A A B P( A or B ) P( A ) P(B ) P(A) + P(B) < 1 S is a sample space. Events A and B are mutually exclusive or disjoint events. S A B P(A|B)=0 P(B|A)=0 Not Mutually exclusive or disjoint events, the events overlap. P( A and B ) 0 A and B S is a sample space. Events A and B are mutually exclusive or disjoint events. P( A or B ) P( A ) P(B ) P( A and B ) Note: P(A) + P(B) does not have to be <1 For independent events: P(A and B) = P(A)P(B) For non-independent events P(A and B) = P(A)P(B|A) P( A and B ) And P( A | B ) P( B ) Problem 1. 1. P(A) = 0.7, P(B) = 0.4, P(A and B) = 0.3 a) Are A and B mutually exclusive? No, because P(A and B) 0 b) Are A and B independent events? No, because P(A and B) P(A)P(B), that is, 0.3 (0.7)(0.4) c) Find P(A or B) P(A or B) = P(A) + P(B) – P(A and B) = 0.7 + 0.4 – 0.3 = 0.8 d) Find P(A|B) P( A and B) 0.3 P( A | B ) 0.75 P( B ) 0.4 Problem 2. 1. P(A) = 0.7, P(B) = 0.4, P(A and B) = 0.28 a) Are A and B mutually exclusive? No, because P(A and B) 0 b) Are A and B independent events? Yes, because P(A and B) =P(A)P(B), that is, 0.28 (0.7)(0.4) c) Find P(A or B) P(A or B) = P(A) + P(B) – P(A and B) = 0.7 + 0.4 – 0.28 = 0.82 d) Find P(A|B) P( A and B ) 0.28 P( A | B ) 0.7 P( A ) P( B ) 0.4 Problem 3. Only two students in the class are independently trying to solve a very difficult probability problem. Student A has a probability of 0.9 of solving the problem while student B has a probability of 0.6 of solving the problem. a) What is the probability that both of them will solve the problem? Solution: Because they are working independently, P(A and B) = P(A)P(B) = (0.9)(0.6) = 0.54 b) What is the probability that both of them fail to solve the problem? Solution: P(A fails to solve the problem and B fails to solve the problem) P( A and B ) P( A )( P(B ) =(1 - 0.9)(1 - 0.6) = (0.1)(0.4) = 0.04 d) What is the probability that the problem will be solved? Solution: The problem is solve if either one or both student solve the problem. P(A or B) = P(A) + P(B) - P (A and B) = 0.9 + 0.6 – 054 = 0.96 d) What is the probability that only A solves the problem? Solution: P(only A) = P(A and B’) = P(A)P(B’) = (0.9)(0.4) = 0.36 e) What is the probability that only one of the students solves the problem? Solution: P(only one) = P(only A or only B) = P(A and B’) + P(A’ and B) = (0.9)(0.4) + (0.1)(0.5) = 0.36 + 0.05 = 0.41