1 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 3-4 Multiplication Rule: Basics

2 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding the Probability of Two or More Selections  Multiple selections  Multiplication Rule

3 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Notation P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)

4 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Ta Tb Tc Td Te Fa Fb Fc Fd Fe abcdeabcdeabcdeabcde TFTF FIGURE 3-9 Tree Diagram of Test Answers

5 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Ta Tb Tc Td Te Fa Fb Fc Fd Fe abcdeabcdeabcdeabcde TFTF FIGURE 3-9 Tree Diagram of Test Answers

6 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Ta Tb Tc Td Te Fa Fb Fc Fd Fe abcdeabcdeabcdeabcde TFTF P(T) = FIGURE 3-9 Tree Diagram of Test Answers 1 2

7 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Ta Tb Tc Td Te Fa Fb Fc Fd Fe abcdeabcdeabcdeabcde TFTF P(T) = P(c) = FIGURE 3-9 Tree Diagram of Test Answers

8 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Ta Tb Tc Td Te Fa Fb Fc Fd Fe abcdeabcdeabcdeabcde TFTF P(T) = P(c) = P(T and c) = FIGURE 3-9 Tree Diagram of Test Answers

9 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P (both correct)

10 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P (both correct) = P (T and c)

11 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P (both correct) = P (T and c)

12 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P (both correct) = P (T and c) = Multiplication Rule

13 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P (both correct) = P (T and c) = Multiplication Rule INDEPENDENT EVENTS

14 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B A as “B given A”). Notation for Conditional Probability

15 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definitions  Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other.  Dependent Events If A and B are not independent, they are said to be dependent.

16 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Formal Multiplication Rule  P(A and B) = P(A) P(B A)  If A and B are independent events, P(B A) is really the same as P(B)

17 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 3-10 Applying the Multiplication Rule P(A or B) Multiplication Rule Are A and B independent ? P(A and B) = P(A) P(B A) P(A and B) = P(A) P(B) Yes No

18 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Intuitive Multiplication When finding the probability that event A occurs in one trial and B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A.

19 Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Small Samples from Large Populations If a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so they are technically dependent).