1. Chapter 4: Probability (Cont.) In this handout: Venn diagrams Event relations Laws of probability Conditional probability Independence of events

2. Example: Toss a coin twice. Let event A corresponds to “ tail at the second toss ”; event B corresponds to “ at least one head ”. Venn Diagram: representing events graphically

3. Example: Two monkeys to be selected by lottery for an experiment. Label the possible pairs (elementary outcomes): {1, 2} e 1 {2, 3} e 4 {1, 3} e 2 {2, 4} e 5 {1, 4} e 3 {3, 4} e 6 Let A: selected monkeys are of the same type; B: selected monkeys are of the same age;

4. The complement of an event A is the set of all elementary outcomes that are not in A. The union of events A and B is the set of all elementary outcomes that are in A, B, or both. The intersection of events A and B is the set of all elementary outcomes that are in A and B.

6. Example: Equal chances that the answer to a problem is correct or wrong. What is the probability of getting at least one correct answer? P(at least one correct answer) = 1 – P(all answers wrong) = 1 – 1/8 = 7/8

7. The probability of an event A must often be modified after information is obtained as to whether or not a related event B has taken place. Example: Conditional probability Q1: Probability that a randomly-selected person has hypertension? Q2: A randomly-selected person is overweight. What is the probability that the person also has hypertension? Let A denote “has hypertension”, B denote “overweight”. Then P( has hypertension given that overweight ) = P( A | B ) =.1/.25 =.4

8. Box on Page 143 Conditioned probability; multiplication law of probability Conditional probability

9. Independence of Events

10. Example: A mechanical system consists of two components. Component 1 has reliability (probability of not failing).98 and component 2 has reliability.95. If the system can function only if both components function, what is the reliability of the system? Let A 1 denote “component 1 functions”, A 2 denote “component 2 functions”, S denote “system functions”. Given that the components operate independently, we take the events A 1 and A 2 to be independent. Thus, P(S) = P(A 1 ) P(A 2 ) =.98 *.95 =.931