Addition Rule and Multiplication Rule

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Chapter 4: Probability
4.2: Addition Rule and Multiplication Rule

1. Elementary Statistics Chapter 4: Probability 4.2 Addition & Multiplication Rules 1

2. Chapter 4: Probability 4.1 Basic Concepts of Probability 4.2 Addition Rule and Multiplication Rule 4.3 Complements and Conditional Probability, and Bayes’ Theorem 4.4 Counting 4.5 Probabilities Through Simulations (available online) 2 Objectives: Determine sample spaces and find the probability of an event, using classical probability or empirical probability. Find the probability of compound events, using the addition rules. Find the probability of compound events, using the multiplication rules. Find the conditional probability of an event. Find the total number of outcomes in a sequence of events, using the fundamental counting rule. Find the number of ways that r objects can be selected from n objects, using the permutation rule. Find the number of ways that r objects can be selected from n objects without regard to order, using the combination rule. Find the probability of an event, using the counting rules.

3. Addition rule: A tool to find P ( A or B ), which is the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of a procedure. The word “ OR ” in the Addition rule is associated with the addition of probabilities. Addition Rule (Union): P ( A or B ) = P (in a single trial, event A occurs or event B occurs or they both occur) To find P ( A or B ), add the number of ways event A can occur and the number of ways event B can occur, but add in such a way that every outcome is counted only once. P ( A or B ) is equal to that sum, divided by the total number of outcomes in the sample space. P ( A or B ) = P ( A ) + P ( B ) − P ( A and B ) where P ( A and B ) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure. 4.2 Addition Rule and Multiplication Rule Key Concept   Complementary Events: Rules 3

4. Disjoint (or mutually exclusive): Events A and B are disjoint (or mutually exclusive ) if they cannot occur at the same time. (That is, disjoint events do not overlap.) P (A & B) = 0 must be disjoint. Multiplication rule: A tool to find P ( A and B ), which is the probability that event A occurs and event B occurs. The word “and” in the multiplication rule is associated with the multiplication of probabilities. Compound Event: A compound event is any event combining two or more simple events.   4.2 Addition Rule and Multiplication Rule Key Concept Venn Diagram for Events That Are Not Disjoint Venn Diagram for Disjoint Events 4

5. Example 1 Determine which events are mutually exclusive (disjoint) and which are not, when a single die is rolled. a. Getting an odd number and getting an even number b. Getting a 3 and getting an odd number c. Getting an odd number and getting a number less than 4 d. Getting a number greater than 4 and getting a number less than 4 5 Mutually Exclusive (Disjoint) a. Getting an odd number: 1, 3, or 5 Getting an even number: 2, 4, or 6 Mutually Exclusive b. Not Mutually Exclusive c. Not Mutually Exclusive d. Getting a number greater than 4: 5 or 6 Getting a number less than 4: 1, 2, or 3 Mutually Exclusive

6. Assume the probability of randomly selecting someone who has sleepwalked is 0.27, so P (sleepwalked) = 0.27. If a person is randomly selected, find the probability of getting someone who has not sleepwalked. 6 P (has not sleepwalked) = 1 − P (sleepwalked) Example 2 Complementary Events: Rules    

7. Example 3 In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male. 7 Addition Rules   Staff Females Males Total Nurses Physicians 8 5   7 1 3 2 Total 10 3 13   P ( A or B ) = P ( A ) + P ( B ) − P ( A and B )

8. P ( A and B ) = P (event A occurs in a first trial and event B occurs in a second trial) P ( B | A ) represents the probability of event B occurring after it is assumed that event A has already occurred. To find the probability that event A occurs in one trial and event B occurs in another trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B is found by assuming that event A has already occurred. Independence and the Multiplication Rule Independent: Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. (Several events are independent if the occurrence of any does not affect the probabilities of the occurrence of the others.) If A and B are not independent, they are said to be dependent. Sampling: In the world of statistics, sampling methods are critically important, and the following relationships hold: Sampling with replacement: Selections are independent events. Sampling without replacement: Selections are dependent events. P ( A ∩ B ) = P ( A )  P ( B ) P ( A and B ) = P ( A )  P ( B | A ) Multiplication Rule 8

9. Example 4 50 test results from the subjects who use drugs are shown below: a. If 2 of these 50 subjects are randomly selected with replacement, find the probability the first selected person had a positive test result and the second selected person had a negative test result. b. Repeat part (a) by assuming that the two subjects are selected without replacement. 9 Multiplication Rules Tested Frequency + 45 − 5 Total 50 P (1st selection is positive and 2nd is negative) Formal Multiplication Rule P ( A and B ) = P ( A )  P ( B | A ) P ( A ∩ B ) = P ( A )  P ( B )   P (1st selection is positive and 2nd is negative) P ( A ∩ B ) = P ( A )  P ( B | A )      

10. Treating Dependent Events and Independent 5% Guideline for Long & Difficult Calculations: When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent (even though they are actually dependent). Example 5: Three adults are randomly selected without replacement from the 247,436,830 adults in the United States. Also assume that 10% of adults in the United States use drugs. Find the probability that the three selected adults all use drugs. Without replacement: The three events are dependent . The sample size of 3 < 5% (247,436,830) ⇾ Assume independent P (all 3 adults use drugs) = P (first uses drugs) · P (second uses drugs) · P (third uses drugs) = (0.10)(0.10)(0.10) = 0.001 10 Given: P (D) = 0.10

11. Redundancy: Important Application of the Multiplication Rule The principle of redundancy is used to increase the reliability of many systems. Our eyes have passive redundancy in the sense that if one of them fails, we continue to see. An important finding of modern biology is that genes in an organism can often work in place of each other. Engineers often design redundant components so that the whole system will not fail because of the failure of a single component. Modern aircraft are now highly reliable, and one factor contributing to that reliability is the use of redundancy, whereby critical components are duplicated so that if one fails, the other will work. For example, the Airbus 310 twin-engine airliner has three independent hydraulic systems (to move and activate landing gear, flaps and brakes) , so if any one system fails, full flight control is maintained with another functioning system. 11

12. F or a typical flight, the probability of an aircraft’s engine hydraulic system for failure is 0.002. If the aircraft had only one hydraulic system, what is the probability that the aircraft’s flight control would work for a flight? An aircraft has 3 independent hydraulic systems, what is the probability that on a typical flight, control can be maintained with a working hydraulic system? Example 5 P (1 hydraulic system does not fail ) = 1 − P (failure) = 1 − 0.002 = 0.998 b. With 3 independent hydraulic systems , flight control will be maintained if the three systems do not all fail. P (All 3 failing) = 0.002 · 0.002 · 0.002 = 0.000000008. It follows that the probability of maintaining flight control is as follows: P (Not All 3 failing) = 1 − 0.000000008 = 0.999999992 12

13. Simulation ( Time ) 13 Solution 1: Flipping a fair coin: heads = female & tails = male H H T H T T H H H H  female female male female male male male female female female Solution 2: Generating 0’s and 1’s with a computer or calculator: 0 = male & 1 = Female 0 0 1 0 1 1 1 0 0 0  male male female male female female female male male male A simulation of a procedure is a process that behaves the same way as the procedure, so that similar results are produced. In many experiments, random numbers are used in the simulation naturally occurring events. Below are some ways to generate random numbers. Gender Selection When testing techniques of gender selection, medical researchers need to know probability values of different outcomes, such as the probability of getting at least 60 girls among 100 children. Assuming that male and female births are equally likely, describe a simulation that results in genders of 100 newborn babies.