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Section 3.3 Addition Rule (Venn Diagram) Section 3.3 Objectives • Determine if two events are mutually exclusive • Use a Venn Diagram to find the probability of two events (uses the Addition Rule) Mutually Exclusive Events Mutually exclusive • Two events A and B cannot occur at the same time A B A and B are mutually exclusive A B A and B are not mutually exclusive Example: Mutually Exclusive Events Decide if the events are mutually exclusive. Event A: Roll a 3 on a die. Event B: Roll a 4 on a die. Solution: Mutually exclusive (The first event has one outcome, a 3. The second event also has one outcome, a 4. These outcomes cannot occur at the same time.) Example: Mutually Exclusive Events Decide if the events are mutually exclusive. Event A: Randomly select a male student. Event B: Randomly select a nursing major. Solution: Not mutually exclusive (The student can be a male nursing major.) The Addition Rule Addition rule for the probability of A or B The probability that events A or B will occur for mutually exclusive events A and B is P(A or B) = P(A) + P(B) (Can be extended to any number of mutually exclusive events) Example: Using the Addition Rule (mutually exclusive events) The frequency distribution shows the volume of sales (in dollars) and the number of months in which a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between $75,000 and $124,999 next month? ( 36 months) Sales volume ($) Months 0–24,999 3 25,000–49,999 5 50,000–74,999 6 75,000–99,999 7 100,000–124,999 9 125,000–149,999 2 150,000–174,999 3 175,000–199,999 1 Solution: Using the Addition Rule • A = monthly sales between $75,000 and $99,999 • B = monthly sales between $100,000 and $124,999 • A and B are mutually exclusive P( A or B) P( A) P( B) 7 9 36 36 16 0.444 36 Sales volume ($) Months 0–24,999 3 25,000–49,999 5 50,000–74,999 6 75,000–99,999 7 100,000–124,999 9 125,000–149,999 2 150,000–174,999 3 175,000–199,999 1 Non- Mutually Exclusive Events D – event of being on Dean’s list A – event of being an athlete P(D) = 0.25 P(A) = 0.20 P(A and D) = 0.05 What is Probability of being an Athlete and not being on the Dean’s list? P(A and not D) A and D are ~not~ mutually exclusive (someone can be an athlete and be on Dean’s list!) A D P(A and D) = .05 (5%) A D 5 P(A) = 20 A 15 D 5 P(D) = 25 A 15 D 5 20 Total Probability is 100% A 15 D 5 20 60 P(A and not D) = 15 A 15 D 5 20 60 P(not A) = 20+60 = 80 P(not D) = 15+60 = 75 P(A’) = 1 – P(A) = 100 – 20 = 80 P(D’) = 1 – P(D) = 100 – 25 = 75 A 15 D 5 20 60 P(A and D) = 5 P(A or D) = 15 + 5 + 20 = 40 A 15 D 5 20 60 P(A or D, but not both) = 15 + 20 = 35 P(not A and not D) = 60 A 15 D 5 20 60 P(A and not D) = 15 P(D and not A) = 20 A 15 D 5 20 60 Non- Mutually Exclusive Events M – event of being male E – event of being younger than 18 P(M) = 79% P(E) = 18% P(M and E) = 14% P(M and E) = 14 M E 14 P(M) = 79 P(E) = 18 M 65 E 14 4 Total probability is 100% M 65 E 14 4 17 P(not M) = 4+17 = 21 [ same as P(M’) ] P(not E ) = 65+17 = 82 [ same as P(E’) ] P(M or E) = 83 P(M and not E) = 65 P(E and not M) = 4 P(not M and not E) = 17 P(M or E, but not both) = 69 M 65 E 14 4 17 P(not M) = 4+17 = 31 [ same as P(M’) ] P(not E ) = 65+17 = 82 [ same as P(E’) ] P(M or E) = 83 P(M and not E) = 65 P(E and not M) = 4 P(not M and not E) = 17 P(M or E, but not both) = 69 M 65 E 14 4 17 Section 3.3 Summary • Determined if two events are mutually exclusive • Used the Addition Rule (using Venn diagram) to find the probability of two events