05 Introduction to Probability Part 1

Combination of two or more events additive rule for

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combination of two or more events Additive rule : For any two events A and B P(A or B) = P(A) + P(B) – P(A and B) Always true! Marble example: P(A or B) = 5/9 +3/9 – 1/9 =7/9 24
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25 Probability of a compound event: Two dice example Elementary events = Sum of Two Dice (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) ei Prob 2 3 4 5 6 7 8 9 10 11 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 P(A) = P(sum is even) = 18/36 = 1/2 P(B) = P(sum ≤ 7) = 21/36 = 7/12
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26 Probability of a compound event: Venn Diagram Venn Diagram: sample space and events A: the sum of two dice is even B: the sum does not exceed 7 Sample space A B Elementary events: 2,3, ….. , 10, 11, 12 8, 10, 12 2 4 6 3, 5, 7 9, 11 Probability of A = P(A) Probability of B = P(B) Called “marginal” probabilities P(A) = 1/2 P(B) = 7/12
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Complement of A (Ac or A’ or A) = Not A Probability of all areas which are not part of A 3 B 2 4 6 8 10 12 27 Complement Sample space 5 7 A 9 11 A: the sum is even A’: the sum is not even 1 – ½ = ½ P(A) = 1/2 P(A’) = 1/2 A’ = - = P(A) 1 ) P(A or ) P(A' or ) A P( c ) A P(
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28 Intersection Intersection (A and B) = Area where both A and B are true Probability of A and B = P(A ∩ B) Called “joint” probability Sample space A B 2 4 6 3 5 7 8 10 12 9 11 P(2,4, or 6)=9/36 A: the sum is even B: the sum is  7 B) P(A B A
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Union (A or B) = Area where either A or B are true Probability of the combined area of A and B = P(A U B) 29 Union Sample space A B 2 4 6 3 5 7 8 10 12 9 11 A: the sum is eve B: the sum is  7 A U B B) P(A
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30 Additive rule For two events A and B Sample space B 3 5 7 A 2 4 6 8 10 12 2 4 6 P(A)=P(2,4,6,8,10, or 12)=18/36 P(B)=P(2,3,4,5,6 or 7)=21/36 P(A∩B)=P(2,4, or 6)=9/36 =18/36 + 21/36 – 9/36 = 30/36 A: the sum is eve B: the sum is  7 P(9 or 11)=6/36 9 11 B) P(A P(B) P(A) B) P(A - + = B A B) P(A P(B) P(A) B) P(A - + =
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For two events A and B: Intersection of complements = complement of union 31 Complement of union Sample s pace B 3 5 7 A 2 4 6 8 10 12 A: the sum is even B: the sum is  7 P(9 or 11)=6/36 9 11 P(A’ ∩ B’) = P(A U B)’ = 1 – P(A U B) = 1 – 30/36 = 6/36 A B’ A’ ∩ B’ =(A U B)’
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Example: Phillies baseball! A survey was recently conducted among 200 sports journalists about the outcomes of the Phillies’ upcoming games against Reds and Cardinals. 50 thought Phillies would beat Reds (R) 110 thought Phillies would beat Cardinals (C) 60 thought Phillies would lose both P(R) = 50/200 = 0.25, P(C) = 0.55, P(R’∩ C’) = 0.30
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33 Example: Phillies baseball! Venn Diagram Representation for Phillies baseball Sample space R C P(R and C)=0.10 R: Phillies beating Reds C: Phillies beating Cardinals P(R) = 0.25 P(C) = 0.55 P(R or C)=0.70 P(R’ ∩ C’) = P(R U C)’ = 0.30 30 journalists 20 journalists 90 journalists 60 journalists R C
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a) What is the probability that a respondent said that Phillies would win both games?
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