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Chapter 4 Probability

Chapter 4 Probability. 4-1 Review and Preview 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting 4-7 Probabilities Through Simulations 4-8 Bayes’ Theorem. Objectives.

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Chapter 4 Probability

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  1. Chapter 4Probability 4-1 Review and Preview 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting 4-7 Probabilities Through Simulations 4-8 Bayes’ Theorem

  2. Objectives Students will find the probability of an event occurring “at least one” time:In short, find the probability of at least one of some specified event. Conditional probability:Students will find the probability of an event given that some other event has already occurred.

  3. Complements: The Probability of “At Least One” The complement of an event occurring at least once is the event never happens at all. • “At least one” means “one or more.”

  4. Example What is the compliment of getting at least one defective dvd in a lot of 50 dvds.

  5. Finding the Probability of “At Least One” To find the probability of at least one of something, calculate the probability of none and then subtract that result from 1. That is, P(at least one) = 1 – P(none).

  6. Steps 1.Let A = getting at least one of some event. 2. A comp = getting none of that event. 3. Find P(A comp) 4. Find P(A) by finding 1 – P(A comp)= P(A).

  7. Example Topford supplies X-Data DVDs in lots of 50, and they have a reported defect rate of 0.5% so the probability of a disk being defective is 0.005. It follows that the probability of a disk being good is 0.995. What is the probability of getting at least one defective disk in a lot of 50? 1.Let A = getting at least one of some event. 2. A comp = getting none of that event. 3. Find P(A comp) 4. Find P(A) by finding 1 – P(A comp)= P(A).

  8. 1.Let A = the event of getting at least one defective disk in a lot of 50. 2. A compliment = getting 50 good disks. 3. Find P(A compliment)= P(good AND good AND good…) 4. Find P(A) by using 1 – P(A comp)= P(A).

  9. Example – continued What is the probability of getting at least one defective disk in a lot of 50?

  10. Conditional Probability A conditional probability, , is the probability of an event B occurring given that some other event A has already occurred. It can be found by dividing the probability of events A and B both occurring together by the probability of event A:

  11. Intuitive Approach to Conditional Probability The conditional probability of B given A can be found by assuming that event A has occurred and then calculating the probability that event B will occur.

  12. Example Refer to the table to find the probability that a subject actually uses drugs, given that he or she had a positive test result.

  13. Example - continued

  14. Note It is incorrect to believe that and are the same. To incorrectly use one value for the other, is often called confusion of the inverse.

  15. Example Refer to the table to find the probability that a subject had a positive test result given that he or she actually uses drugs.

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