1 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 3-5 Multiplication Rule: Complements and Conditional Probability

2 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of ‘At Least One’  ‘At least one’ is equivalent to ‘one or more’.

3 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of ‘At Least One’  ‘At least one’ is equivalent to ‘one or more’.  The complement of getting at least one item of a particular type is that you get no items of that type.

4 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of ‘At Least One’  ‘At least one’ is equivalent to ‘one or more’.  The complement of getting at least one item of a particular type is that you get no items of that type. IfP(A) = P(getting at least one), then P(A) = 1 - P(A) where P(A) is P(getting none)

5 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of ‘At Least One’  Find the probablility of a couple have at least 1 girl among 3 children.

6 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of ‘At Least One’  Find the probablility of a couple have at least 1 girl among 3 children. IfP(A) = P(getting at least 1 girl), then P(A) = 1 - P(A) where P(A) is P(getting no girls)

7 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of ‘At Least One’  Find the probablility of a couple have at least 1 girl among 3 children. IfP(A) = P(getting at least 1 girl), then P(A) = 1 - P(A) where P(A) is P(getting no girls) P(A) = (0.5)(0.5)(0.5) = 0.125

8 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of ‘At Least One’  Find the probablility of a couple have at least 1 girl among 3 children. IfP(A) = P(getting at least 1 girl), then P(A) = 1 - P(A) where P(A) is P(getting no girls) P(A) = (0.5)(0.5)(0.5) = P(A) = = 0.875

9 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Conditional Probability Definition The conditional probability of event B occurring, given that A has already occurred, can be found by dividing the probability of events A and B both occurring by the probability of event A.

10 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Conditional Probability P(A and B) = P(A) P(B | A)  Formal P(B | A) = P(A and B) P(A)

11 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Conditional Probability P(A and B) = P(A) P(B | A)  Formal P(B | A) =  Intuitive P(A and B) P(A)

12 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Conditional Probability P(A and B) = P(A) P(B | A)  Formal P(B | A) =  Intuitive The conditional probability of B given A can be found by assuming the event A has occurred and, operating under that assumption, calculating the probability that event B will occur. P(A and B) P(A)

13 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman If P(B | A) = P(B) then the occurrence of A has no effect on the probability of event B; that is, A and B are independent events. Testing for Independence

14 Chapter 3. Section 3-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman If P(B | A) = P(B) then the occurrence of A has no effect on the probability of event B; that is, A and B are independent events. or If P(A and B) = P(A) P(B) then A and B are independent events. Testing for Independence