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Chapter 4

Chapter 4. Probability Concepts. 4.1 Events and Probability. Three Helpful Concepts in Understanding Probability: Experiment Sample Space Event Experiment An activity for which the outcome is uncertain is an experiment. Example 4.1.1: Examples of experiments Flipping a coin

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Chapter 4

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  1. Chapter 4 Probability Concepts

  2. 4.1 Events and Probability • Three Helpful Concepts in Understanding Probability: • Experiment • Sample Space • Event • Experiment • An activity for which the outcome is uncertain is an experiment. • Example 4.1.1: Examples of experiments • Flipping a coin • Rolling two dice • Taking an exam • Observing the number of arrivals at a drive-up window over a 5-minute period

  3. 4.1 Events and Probability (cont.) • Sample Space • The list of all possible outcomes of an experiment is called the sample space. • Example 4.1.2: Example of sample space • Flipping a coin twice results in one of four possible outcomes. These possible outcomes are HH, HT, HT, TT. Therefore, sample space = {HH, HT, TH, TT}. • If there are n outcomes of an experiment, sample space lists all n outcomes.

  4. 4.1 Events and Probability (cont.) • Event • An event consists of one or more possible outcomes of the experiment. • It is usually denoted by a capital letter. • Example 4.1.3: Examples of experiments and some corresponding events • Experiment: Rolling two dice; events: A = rolling a total of 7, B = rolling a total greater than 8, C = rolling two 4s. • Experiment: Taking an exam; events: A = pass, B = fail. • Experiment: Observing the number of arrivals at a drive-up window over a 5-minute period; events: A0 = no arrivals, A1 = seven arrivals, etc.

  5. 4.1 Events and Probability (cont.) • Probability • A numerical measure of the chance OR likelihood that a particular event will occur. • The probability that event A will occur is written P(A). • The probability of any event ranges from 0 to 1, inclusive. • P(A) = 0 means event A will never occur. • P(A) = 1 means event A must occur.

  6. 4.1 Events and Probability (cont.) • How to come up with probability? • Classical Definition of Probability • If event A occurs in m of the n outcomes in an experiment, then the probability that event A will occur is: • This assumes all n possible outcomes have an equal chance of occurring. • Example 4.1.4: Toss a nickel and a dime. The sample space (i.e., the list of the possible outcomes) is {HH, HT, TH, TT}. If event A is observing one head and one tail, then m = 2 and n = 4. So according to classical definition of probability, P(A) = m/n = 2/4 = 0.5.

  7. 4.1 Events and Probability (cont.) • Relative Frequency Approach • Observe an experiment n times and count the number of times event A occurs, m. • Example 4.1.5: A production process has been in operation in for 250 days and has been accident-free for 220 days. If event A is a randomly chosen accident-free day in the future, then, according to relative frequency approach, P(A) = 220/250 = 0.88. • Subjective Probability • A measure (between 0 and 1) of your belief that a particular event will occur. • Example 4.1.6: Example of subjective probability: The probability that it will rain today is 50%.

  8. 4.2 Basic Concepts • Contingency Table (also called Cross-Tab Table) • Contingency tables are used to record and analyze the relationship between two variables. • Example 4.2.1: Datacomp Survey: Datacomp recently conducted a survey of 200 selected purchasers of their newly introduced laptop computer to obtain a gender-and-age profile of its new customers. The data are summarized in the following contingency table. Some Events: M = a male is selected B = the person selected is between 30 & 45 F = a female is selected O = the person selected is over 45 U = the person selected is under 30

  9. 4.2 Basic Concepts (cont.) • Contingency Table (also called Cross-Tab Table) • Example 4.2.2: At a local University 75% of the Business faculty are professors and 70% of the faculty are full time. 80% of the professors are full time. Suppose the faculties are randomly assigned to courses. • If you take a course in the Business School, what is the probability that you will get a Professor for the course? • What is the probability that a teacher selected at random is a ProfessorAND is Full Time? • What is the probability that a teacher chosen at random is Not a Professor OR is Not Full Time?

  10. 4.2 Basic Concepts (cont.) • Marginal Probability • Marginal probability is the probability of one event, regardless of the other events. • Example 4.2.2: In Datacomp Survey, the marginal probabilities are: • P(M) = 120/200 = 0.6 • P(F) = 80/200 = 0.4 • P(U) = 0.5 • P(B) = 0.25 • P(O) = 0.25

  11. 4.2 Basic Concepts (cont.) • Complement of an event • The complement of an event A is the event that A does not occur. • This event is denoted by A. • For example, A = it rains tomorrow, A = it does not rain tomorrow. • Example 4.2.3: In Datacomp Survey: M = a male is selected. M = a male is not selected = a female is selected. P(M) = 0.6, and so P(M) = P(F) = 0.4. • P(A) + P(A) = 1 • P(A) = 1 – P(A) • P(A) = 1 – P(A)

  12. 4.2 Basic Concepts (cont.) • Joint Probability • The probability of the occurrence of two events at the same time • Example 4.2.4: In Datacomp Survey, what proportions are males between 30 and 45? That is, find the probability of selecting a person who is a male and between 30 and 45. P(M and B) = 20/200 = 0.10 • Example 4.2.5: The probability of selecting a person who is a female and under 30 is P(F and U) = 40/200 = 0.20.

  13. 4.2 Basic Concepts (cont.) • Either of Two Events • The probability of either event A or event B occurring is written as P(A or B). • Example 4.2.6: In Datacomp Survey, the probability of selecting a person who is male or under 30 is P(M or U) = (120 + 40) / 200 = 0.80. • Conditional Probability • Whenever you are given information and are asked to find a probability based on this information, the result is a conditional probability. • This probability is written as P(A|B) and read as “probability of A given B”. • Example 4.2.7: In Datacomp Survey, what is the probability that a randomly selected customer is male given that he is under 30? P(M | U) = 60/100 = 0.60.

  14. 4.2 Basic Concepts (cont.) • Independent Events • Events A and B are independent if the probability of event A is unaffected by the occurrence or nonoccurrence of event B. • Events A and B are independent if and only if: • P(A | B) = P(A) (assuming P(B) ≠ 0), or • P(B | A) = P(B) (assuming P(A) ≠ 0), or • P(A and B) = P(A) • P(B) • Example 4.2.8: In Datacomp Survey, are events M and U independent? • P(M) = 120/200 = 0.6, P(M | U) = 60/100 = 0.6, so they are independent. • P(U) = 100/200 = 0.5, P(U | M) = 60/120 = 0.5, so they are independent. • P(M and U) = 60/200 = 0.3, P(M) • P(U) = 0.6 • 0.5 = 0.3, so they are independent.

  15. 4.2 Basic Concepts (cont.) • Dependent Events • Events that are not independent are dependent events. • P(A | B) =P(A and B) /P(B) • P(B | A) =P(A and B) /P(A) • P(A and B) = P(A | B) • P(B) = P(B | A) • P(A)

  16. 4.2 Basic Concepts (cont.) • Mutually Exclusive Events • If an event can not occur when another event has occurred the two events are said to be mutually exclusive. • Events A and B are mutually exclusive if their joint probability is zero, that is, P(A and B) = 0. • P(A or B) = P(A) + P(B) • Example 4.2.9: Consider an experiment of randomly selecting a card fro a deck of 52 cards. Is the event of selecting a queen mutually exclusive from the event of selecting a heart? No. Why? • Non-mutually Exclusive Events • P(A and B) ≠ 0. • P(A or B) = P(A) + P(B) - P(A and B).

  17. B A Venn diagram for events A and B. The rectangle represents all possible outcomes of an experiment 4.3 Going Beyond the Contingency Table • Venn Diagram • In Venn diagram, a rectangle represents all possible outcomes of an experiment. • Event are shown in the rectangle as circles. • The probability of an event occurring is its corresponding area in the Venn diagram. A 0.6 A 0.4 Venn diagram for P(A) = 0.4.

  18. A B A B 4.3 Going Beyond the Contingency Table (cont.) Venn diagram for P(A and B). The points in the shaded area are in A and B Venn diagram for P(A or B). The points in the shaded area are in A or B A Venn diagram of mutually exclusive events. P(A and B) = 0 and P(A or B) = P(A) + P(B). B

  19. 4.3 Going Beyond the Contingency Table (cont.) • Probability Rules • General Additive Rule P(A or B) = P(A) + P(B) - P(A and B) • Special Additive Rule If A and B are mutually exclusive then P(A or B) = P(A) + P(B) • Example 4.3.1: In Datacomp Survey, what is the probability of selecting a person who is male or under 30? That is, find P(M or U). P(M or U) = P(M) + P(U) – P(M and U) = 120/200 + 100/200 – 60/200 = 0.80 • Example 4.3.2: The probability of event A is 0.5 and the probability of event B is 0.2. If P(A and B) is 0.1, what is P(A or B)? P(A or B) = P(A) + P(B) – P(A and B) = 0.5 + 0.2 – 0.1 = 0.6 • Example 4.3.3: In Datacomp Survey, find P(M or F). P(M or F) = P(M) + P(F) – P(M and F) = 120/200 + 80/200 – 0/200 = 0.6 + 0.4 – 0 = 0.6 + 0.4 = P(M) + P(F) = 1.0

  20. 4.3 Going Beyond the Contingency Table (cont.) • General Conditional Probability Rule • Special Conditional Probability Rule If events A and B are independent then:

  21. 4.3 Going Beyond the Contingency Table (cont.) • Example 4.3.4: If P(A and B) = 0.4 and P(B) = 0.8, find P(A|B). • Example 4.3.5: If P(A) = 0.3 and P(B) = 0.4, and P(A and B) = 0.2, are events A and B statistically independent? Use conditional probability rules. Events A and B are not independent.

  22. 4.3 Going Beyond the Contingency Table (cont.) • Multiplicative Rule • Special Multiplicative Rule If events A and B are independent then: • Example 4.3.6: Let P(A) = 0.6, P(B) = 0.2, and P(A|B) = 0.1. Find P(A and B)

  23. 4.3 Going Beyond the Contingency Table (cont.) • Sampling Without Replacement • Assume that you select a card from a deck, examine it, and then discard it. You then select another card. This procedure is called sampling without replacement. • Example 4.3.7: Let A = selecting a king on the first draw, and B = selecting a king on the second draw. What is the probability of drawing two kings [P(A and B)]? If you select a king on the first draw, then, of the 51 cards remaining, three are kings. So, P(A) = 4/52 and P(B|A) = 3/51. P(A and B) = (4/52)(3/51) = 0.0045.

  24. 4.3 Going Beyond the Contingency Table (cont.) • Sampling With Replacement • Assume that you select a card from a deck and replace it before selecting the second card. This procedure is called sampling with replacement. • Example 4.3.8: Let A = selecting a king on the first draw, and B = selecting a king on the second draw. What is P(B|A)? There are still 52 cards in the deck. So, P(B|A) = 4/52. • Using Excel to Construct a Contingency Table • KPK Data Analysis > Qualitative Data Charts > Contingency Table.

  25. E1 B E2 B . . . En B 4.4 Tree Diagrams • A tree diagram shows all possible outcomes of an experiment and the probabilities of each. • A general form of a tree diagram is: • Example 4.4.1: Draw a tree diagram for the Datacomp Survey data.

  26. 4.4 Tree Diagrams (cont.) • Rules for Tree Diagram • Rule # 1: The probability of the event on the right side (say, event B) of the tree is equal to the sum of the paths; that is, all probabilities along a path leading to event B are multiplied, and then summed over all paths leading to B. • Rule # 2: The posterior probability for the ith path is:

  27. Shift 1 (.06) Defective (.5) 2 (.2) (.08) Defective 3 (.3) (.15) Defective 4.4 Tree Diagrams (cont.) • Example 4.4.2: Zetadyne Corporation 50% of components produced on shift 1 20% of components produced on shift 2 30% of components produced on shift 3 6% of the components produced on shift 1 are defective 8% of the components produced on shift 2 are defective 15% of the components produced on shift 3 are defective

  28. third path sum of paths P(shift 3 | defective) = = = = .495 (.3)(.15) .091 (.045) .091 4.4 Tree Diagrams (cont.) Solution 1 – What percentage of the components are defective? P(Defective) = sum of paths = (.5)(.06) + (.2)(.08) + (.3)(.15) = .030 + .016 + .045 = .091 Solution 2 – Given that a defective component is found, what is the probability that it was produced during shift 3?

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