Chapter 4 Probability

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. A bag contains 8 red marbles, 4 blue marbles, and 1 green marble. Find P(not blue). • A company manufactures glass bottles. The probability that a randomly selected bottle is defective is 0.002. What is the probability that a randomly selected bottle is not defective?

3. Objective Students will use the multiplication rule used for finding P(A and B). Students will determine if two events are dependent or independent. In necessary cases students will use the adjusted multiplication rule for P(A and B) when events are dependent such as sampling without replacement.

4. P(A and B) P(A and B) = the probability that event A occurs and event B occurs. Where event A occurs in the first trial and event B occurs in the second trial. The key word in this section is “and.” Just as the word or is associated with addition, the word and is associated with multiplication.

5. P(B | A) P(B | A) is read as “the probability of B given A” P(B | A) represents the probability of event B occurring given that event A has already occurred. This is especially important when the occurrence of A affects the probability of B.

6. Formal Multiplication Rule This is always true.

7. Intuitive Multiplication Rule When finding the probability that event A occurs in one trial and event 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.

8. Dependent and 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 similarly 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.

9. Note about dependent events Two events are dependent if the occurrence of one of them affects the probability of the occurrence of the other. This does not necessarily mean that one of the events is a cause of the other.

10. Treating Dependent Events as Independent Some calculations are cumbersome, but they can be made manageable by using the common practice of treating events as independent when small samples are drawn from large populations. In such cases, it is rare to select the same item twice. Specifically, when the sample is less than 5% of the population we can treat all events as independent.

11. 1. Find P(A)2. Find P(B|A) note that for independent events this is just P(B). 3. Multiply P(A)P(B|A) 4. This is P(A and B) Algorithm for finding P(A and B)

12. Applying the Multiplication Rule

13. Sampling with replacement Sampling without replacement When finding the probability of more than one event I must consider if I am sampling with or without replacement.

14. Example Find the probability P(a jack and a king) • Sampling with replacement. • Sampling without replacement. a) 1/13 x 1/13 = 1/169 with replacement. b) 1/13 x 4/51 = 4/663 without replacement.

15. Caution When applying the multiplication rule, always consider whether the events are independent or dependent, and adjust the calculations accordingly.

16. Example Suppose 50 drug test results are given from people who use drugs: Find the probability P(a positive result and a negative result) If 2 of the 50 subjects are randomly selected a)With replacement b) Without replacement

17. Example – continued If 2 of the 50 subjects are randomly selected with replacement, find the probability that the first person tested positive and the second person tested negative. Since the process is done with replacement the events are independent and P(A and B) is just P(A)P(B)= 44/50x6/50 = 264/2500 = 0.1056

18. Example – continued If 2 of the 50 subjects are randomly selected without replacement, find the probability that the first person tested positive and the second person tested negative.

19. Example Airport baggage scales can show if bags are overweight to impose high additional fees in some cases. The New York City Department of Consumer Affairs checked all 810 scales at JFK and LaGuardia and 102 scales were found to be defective and ordered out of use. (Based on data reported in The New York Times.) a) If 2 of the 810 scales are randomly selected with replacement find the probability that they are both defective. b) If 2 of the 810 scales are randomly selected without replacement find the probability that they are both defective. c) A larger population of 10,000 scales includes exactly 1,259 defective ones. If 5 scales are randomly selected without replacement, find the probability that they all defective.

20. Example • Airport baggage scales can show if bags are overweight to impose high additional fees in some cases. The New York City Department of Consumer Affairs checked all 810 scales at JFK and LaGuardia and 102 scales were found to be defective and ordered out of use. (Based on data reported in The New York Times.) • If 2 of the 810 scales are randomly selected with replacement find the probability that they are both defective. • 102/810 x 102/810 = 0.0159 • b) If 2 of the 810 scales are randomly selected without replacement find the probability that they are both defective. • 102/810 x 102/809 = 0.0157 • c) A larger population of 10,000 scales includes exactly 1,259 defective ones. If 5 scales are randomly selected without replacement, find the probability that they all defective. • 1,259/10,000 x 1,259/9,999 x 1,259/9,998 x 1,259/9,997 x 1,259/9,996=0.0000314

22. Multiplication Rule for Several Events In general, the probability of any sequence of independent events is simply the product of their corresponding probabilities.

23. Example When two different people are randomly selected from those in your class, find the indicated probability by assuming birthdays occur on the same day of the week with equal frequencies. Probability that two people are born on the same day of the week. Probability that two people are both born on Monday.

24. Example – continued Probability that two people are born on the same day of the week. Because no particular day is specified, the first person can be born on any day. This implies a probability of 1. The probability that the second person is born on the same day is 1/7, so the probability both are born on the same day is 1x1/7=1/7. Probability that two people are both born on Monday. The probability the first person is born on Monday is 1/7, and the same goes for the second person. The probability they are both born on Monday is: .

25. Example When two different people are randomly selected from those in your class, find the indicated probability by assuming birthdays occur on the same day of the week with equal frequencies. • Probability that two people are born on the same day of the week. • Probability that two people are both born on Monday.

26. Tree Diagrams A tree diagram is a picture of the possible outcomes of a procedure, shown as line segments emanating from one starting point. These diagrams are sometimes helpful in determining the number of possible outcomes in a sample space, if the number of possibilities is not too large.

27. Tree Diagrams This figure summarizes the possible outcomes for a true/false question followed by a multiple choice question. Note that there are 10 possible combinations.

28. Summary of Fundamentals • In the addition rule, the word “or” in P(A or B) suggests addition. Add P(A) and P(B), being careful to add in such a way that every outcome is counted only once. • In the multiplication rule, the word “and” in P(A and B) suggests multiplication. Multiply P(A) and P(B), but be sure that the probability of event B takes into account the previous occurrence of event A.