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**Aim: What are ‘Or’ Probabilities?**

Do Now: What is the probability of spinning a number greater than 8 or an odd number?

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Probability of A or B What is the probability of spinning a number greater than 8 or an odd number? Count the number of successes for n > 8 n - odd not yet counted 9, 10, 11, 12 4 1, 3, 5, 7 4 > 8 odd = {1, 3, 5, 7, 9, 10, 11, 12}

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Union The union of sets A and B, denoted by A B, is the set consisting of all elements of A or B or both. A B = {x|x A OR x B} U Region II Region IV Reg. III A B Region I The union of sets A and B is region II, II & IV. ‘or’ is the term used to describe union

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Union of Do Now > 8 odd 9 11 10, 12 1, 3, 5, 7, U 2, 4, 6, 8 U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} > 8 = {9, 10, 11, 12} odd = {1, 3, 5, 7, 9, 11} A B = {1, 3, 5, 7, 9, 10, 11, 12} n(A B) = 8

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Independent Events Mutually exclusive – two events A & B are mutually exclusive if they can not occur at the same time. That is, A and B are mutually exclusive when A B = An outcome for A or B is in one or the other. If the events are mutually exclusive the P(A or B) = P(A) + P(B) If one card is randomly selected from a deck of cards, what is the probability of selecting a king or a queen? mutually exclusive? yes

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**‘Or’ Probabilities Not Mutually Exclusive**

From a standard deck you randomly select one card. What is the probability of selecting a diamond or a face card? mutually exclusive? no P(or fcd) = common elements A B n(A B) = 3 {K, Q, J}

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Probability of (A or B) P(A or B) = P(A) + P(B) - P(A and B) P(A B) = P(A) + P(B) - P(A B) P(A B) = n(A) + n(B) - n(A B) n(S) n(S) n(S) If A and B are not mutually exclusive events, then Example: Find the probability of rolling a die and getting a number that is odd or greater than 2. successes {1,3,5} {3,4,5,6}

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**1 3 5 4 6 3 5 2 1 3 5 4 6 Model Problem odd > 2 P(odd) = 3/6**

Find the probability of rolling a die and getting a number that is odd or greater than 2. odd > 2 1 3 5 4 6 3 5 P(odd) = 3/6 P(> 2) = 4/6 2 A B = {1, 3, 4, 5, 6} 1 3 5 4 6 n(A B) = 5 n(U) = 6

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Model Problem In a group of 50 students, 23 take math, 11 take psychology, and 7 take both. If one student is selected at random, find the probability that the student takes math or psychology P(A B) = P(A) + P(B) - P(A B) 23 4 16 7 M Psy

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**1. A card is drawn from a standard deck of 52. Find P(Ace or jack)**

Model Problems 1. A card is drawn from a standard deck of 52. Find P(Ace or jack) 4 52 P(ace) = 4 52 P(jack) = mutually exclusive P(ace or jack) = 4 52 + 8 = 2. A card is drawn from a standard deck of 52. Find P(king or face card) not mutually exclusive 4 52 P(king) = 12 52 P(face) = P(king or face) = 4 52 12 + _ =

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**A red king must be red and a king 2 P(red and king) = 52**

Model Problems In drawing a card from the deck at random, find the probability that the card is: A. A red king B. A 10 or an ace C. A jack or a club P(A and B) = P(A) · P(B) A red king must be red and a king 2 P(red and king) = 52 mutually exclusive P(A B) = P(A) + P(B) - P(A B) 10’s and aces have no common outcomes P(10’s or aces) = 4 52 + _ = 8 not mutually exclusive P(A B) = P(A) + P(B) - P(A B) There are 4 jacks and 13 clubs, but one of the cards is both (jack of clubs) P(jacks or clubs) = 4 52 13 + 1 _ 16 =

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**P(A B) = P(A) + P(B) - P(A B)**

Model Problems Based on the table below, if one person is randomly selected from the US military, find the probability that this person is in the Army or is a woman. P(A B) = P(A) + P(B) - P(A B) not mutually exclusive Active Duty US Military Personnel, in 000’s Air Force Army Marines Navy Total Male 290 400 160 320 1170 Female 70 10 50 200 360 470 170 370 1370 P(Army Female) = P(A) + P(F) - P(A F)

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Probability Rules 1. The probability of an impossible event is 0. 2. The probability of an event that is certain to occur is 1. 3. The probability of an event E must be greater than or equal to 0 and less that or equal to 1. 4. P(A and B) = n(A B) n(S) 5. P(A or B) = P(A) + P(B) - P(A B) 6. P(Not A) = 1 - P(A) 7. The probability of any even is equal to the sum of the probabilities of the singleton outcomes in the event. 8. The sum of the probabilities of all possible singleton outcomes for any sample space must always equal 1.

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Model Problems Five more men than women are riding a bus as passengers. The probability that a man will be the first passenger to leave the bus is 2/3. How many passengers on the bus are men, and how many are women? x = number of women 5 x + 5 = number of men 10 2x + 5 = number of passengers Number of men P(man) = Number of passengers 2 3 x + 5 2x + 5 = 4x = 3x + 15 x = 5 x + 5 = 10

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The Product Rule

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