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05 Introduction to Probability Part 1

# Probability of a compound event recall that an event

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Probability of a compound event Recall that an event is a set of outcomes of interest One or more heads in a two-coin toss At least one red marble in two draws Sometimes, an event is described as a 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 25 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
26 Probability of a compound event: Venn Diagram Venn Diagram: sample space and events 26 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 27 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(
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 28 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 29 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
30 Additive rule 30 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)’
Example: Phillies baseball! A survey was recently conducted among 200

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Probability of a compound event Recall that an event is a...

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