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The Practice of Statistics Third Edition Chapter 6: Probability and Simulation: The Study of Randomness Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

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Two computer simulations of tossing a balanced coin 100 times

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A Head A four An Ace

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Ex. What is the sample space for the roll of a die? s= {1,2,3,4,5,6} All equal probability What is the sample space for the roll of a pair of dice? s={2,3,4,5,6,7,8,9,10,11,12} different probability

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Events, Sample Spaces and Probability An event is a specific collection of sample points: –Event A: Observe an even number on the roll of a single die. Often represented by a venn diagram.

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Tree diagrams help to determine the sample space Ex. An experiment consists of flipping a coin and tossing a die.

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Ex. For your dinner you need to choose from six entrees, eight sides, and five desserts. How many different combinations of entrée, side and dessert are possible? 6x8x5 = 240

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Unions and Intersections Compound Events Made of two or more other events Union A B Either A or B, or both, occur Intersection A B Both A and B occur

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10 Venn diagrams for (a) event (not E), (b) event (A & B), and (c) event (A or B)

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McClave: Statistics, 11th ed. Chapter 3: Probability 12 The Additive Rule and Mutually Exclusive Events Events A and B are mutually exclusive (disjoint) if A B contains no sample points.

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Complementary Events The complement of any event A is the event that A does not occur, A C. A: {Toss an even number} A C : {Toss an odd number} B: {Toss a number ≤ 4} B C : {Toss a number ≥ 5} A B = {1,2,3,4,6} [A B] C = {5} (Neither A nor B occur)

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Complementary Events

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What is the probability of rolling a five when a pair of dice are rolled?

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P(A) = 4/36 =.111 =11.1%

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Independent Events – two events are independent if the occurrence of one event does not change the probability that the other occurs. Ex. What is the probability of rolling a 7 each of three consecutive roles of a pair of dice? P(rolling 7) = 6/36 = 1/6 ; P( three 7s) = (1/6) 3 =0.5%

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If A and B are mutually exclusive,

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Ex. Event A = { a household is prosperous; Inc. > 75 k} Event B = { a household is educated; completed college} P(A) = 0.125P(A and B) = 0.077 P(B) = 0.237 What is the probability the household is either prosperous or educated? P (A or B) = P(A) + P(B) – P(A and B) = 0.125 + 0.237 – 0.077 = 0.285

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Conditional probability - The probability of the occurrence of an event given that another event has occurred. ♣ Notation – P(B|A) > probability of B given A

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Ex. A die has sides 1,2,3 painted red and sides 4,5,6 painted blue. Event A = {Roll Blue}Event B = {Roll Red} Event C = {Roll even}Event D = {Roll Odd} What is the probability that you roll an even given we rolled a blue? What is the probability of rolling a red, even? What is the probability that you roll an odd given you rolled a blue?

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General Multiplication rule can be extended for any number of events. P(A and B and C) = P(A) x P(B|A) x P(C|A and B)

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Example: Deborah and Matthew have applied to become a partner in their law firm. Over lunch one day they discuss the possibility. They estimate that Deborah has a 70% chance of becoming partner while Matthew has a 50% chance. They also believe that there about a 30% chance that both of them will be chosen. a)What is the probability that either Deborah or Matthew become partner? b)What is the probability that either Deborah or Matthew become partner but not both? c)What is the probability that only Mathew becomes partner? d)What is the probability that neither one becomes partner? e)Are the events Independent? a)P(D U M) = P(D) + P(M) – P(D∩M) = 0.7 + 0.5 – 0.3 = 0.9 b)P(D∩M c ) U P(D c ∩M) = P(D∩M c ) + P(D c ∩M) = 0.4 + 0.2 = 0.6 c) P(D c ∩M) = P(M) x P(D c M) = 0.2 d) P(D c ∩M c ) = 0.1 Event D = {Deborah is made partner} Event M = {Matthew is made partner} e)P(D∩M) = P(D) x P(M) if independent 0.3 ≠ 0.7 x 0.5; No, they are not independent

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Example: 29% of internet users are 1-20 years old, 47% are 21- 50 years old and 24% are 51-90 years old. 47% of 1- 20 year old users chat, 21% of 21-50 year old users chat and 7% of 51-90 year old users chat a)What is the probability that a 19 year old internet user does not chat? b)What is the probability that an internet user chats? c)If we know an internet user chats, what is the probability that they are 60 years old? a) P(C c ∩ A 1 ) = P (A 1 ) x P(C c A 1 ) = 0.29 x 0.53 = 0.1537 b) P(C) = P(C ∩ A 1 ) + P(C ∩ A 2 ) + P(C ∩ A 3 ) 0.1363 +0.0987 + 0.0168 = 0.2518 c) P(A 3 C) = P(A 3 ∩ C) / P(C) = 0.0168/(0.1363 +0.0987 + 0.0168) = 0.0667 Event A 1 = {person is 1-20 years old} Event A 2 = {person is 21-50 years old} Event A 3 = {person is 51-90 years old} Event C = {person chats online}

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