Math 370 | Section X Lecture 1 – General Probability 1) Probability Spaces and Events: Probability/Sample Space (S) Events Union of Events (A ∪ B) Intersection of Events (A ∩ B) Mutually Exclusive Events (A ∩ B = ∅) Exhaustive Events (A ∪ B = S) Complement of Event (A′) Subevent (or subset) of Event (A ⊂ B; A ⊄ B) DeMorgan’s Laws: (A ∪ B)′ = A′ ∩ B′ (A ∩ B)′ = A′ ∪ B′ ACBS2) Event Rules: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A = A ∩ (B ∪ B′) (A′)′ = A 3) Probability Rules: P(S) = 1 P(∅) = 0 For any event A, 0 ≤ P(A) ≤ 1 P(A ∪ B) = P(A) + P(B) – P(A ∩ B) For mutually exclusive events, P(A ∪ B) = P(A) + P(B) (Since P(A ∩ B) = 0) ** 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) + P(A′) = 1 If P is a uniform probability function on probability space with k points, and if the event A consists of m of those points, then P(A) = m/kExamples: 1) Suppose that you toss a six-faced die: a) What is the sample space of all possible outcomes? b) Let event A = “an even number is tossed”. What are A’s possible outcomes? c) Let event B = “an odd number is tossed”. What are B’s possible outcomes? d) What is A ∪ B? A ∩ B? e) Are A and B mutually exclusive events? f) Are A and B exhaustive events? g) Let’s assume that each of the six faces has the same probability of occurring when the die is tossed. What is P(A ∪ B)? P(A ∩ B)? 2) Suppose that P(A ∩ B) = .2 , P(A) = .6 , and P(B) = .5 . Find: a) P(A′ ∪ B′) b) P(A′ ∩ B′) c) P(A′ ∩ B) d) P(A′ ∪ B) 3) A survey is made to determine who will emerge as the ruler of the Seven Kingdoms. Out of the US population, it is found that 75% place their faith in Jon Snow (J), 65% in Daenerys Targaryen (D), 55% in Cersei Lannister (C), 50% in (JD), 40% in (JC), 30% in (DC), and 20% in all three. Find the probability that a person has faith in at least one of these rulers. (SPOILER ALERT: STARK SUPPORTERS ARE LOWER THAN YOU EXPECT!) 4) In a survey of 120 students, the following data was obtained. 60 took English, 56 took Math, 42 took Chemistry, 34 took English and Math, 20 took Math and Chemistry, 16 took English and Chemistry, 6 took all three subjects. Find the number of students who took a) None of the subjects b) Math, but not English or Chemistry, c) English and Math but not
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