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. An activity for which the outcome is uncertain is an experiment. Example 4.1.1: Examples of experiments Example 4.1.1: Examples of experiments Flipping a coin Flipping a coin Rolling two dice Rolling two dice Taking an exam Taking an exam Observing the number of arrivals at a drive-up window over a 5-minute period 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. An event consists of one or more possible outcomes of the experiment. It is usually denoted by a capital letter. It is usually denoted by a capital letter. Example 4.1.3: Examples of experiments and some corresponding events 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: 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: 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: A 0 = no arrivals, A 1 = seven arrivals, etc. Experiment: Observing the number of arrivals at a drive-up window over a 5-minute period; events: A 0 = no arrivals, A 1 = 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. 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. 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. 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 = 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 = Subjective Probability Subjective Probability A measure (between 0 and 1) of your belief that a particular event will occur. 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%. 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. Age (Years) < > 45 Sex(U)(B)(O)Total Male (M) Female (F) Total Some Events: M = a male is selectedB = the person selected is between 30 & 45 F = a female is selectedO = 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 Professor AND is Full Time? What is the probability that a teacher chosen at random is Not a Professor OR is Not Full Time?

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(F) = 80/200 = 0.4 P(U) = 0.5 P(U) = 0.5 P(B) = 0.25 P(B) = 0.25 P(O) = 0.25 P(O) = 0.25

Basic Concepts (cont.) Complement of an event Complement of an event The complement of an event A is the event that A does not occur. The complement of an event A is the event that A does not occur. This event is denoted by A. This event is denoted by A. For example, A = it rains tomorrow, A = it does not rain tomorrow. For example, A = it rains tomorrow, A = it does not rain tomorrow. Example 4.2.3: In Datacomp Survey: 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)

Basic Concepts (cont.) Joint Probability The probability of the occurrence of two events at the same time 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. 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 = 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.

Basic Concepts (cont.) Either of Two Events The probability of either event A or event B occurring is written as P(A or B). 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) = ( ) / 200 = Example 4.2.6: In Datacomp Survey, the probability of selecting a person who is male or under 30 is P(M or U) = ( ) / 200 = Conditional Probability Whenever you are given information and are asked to find a probability based on this information, the result is a 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”. 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? 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.

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(A | B) = P(A)(assuming P(B) ≠ 0), or P(B | A) = P(B)(assuming P(A) ≠ 0), or P(B | A) = P(B)(assuming P(A) ≠ 0), or P(A and B) = P(A) P(B) P(A and B) = P(A) P(B) Example 4.2.8: In Datacomp Survey, are events M and U independent? 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(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(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) = = 0.3, so they are independent. P(M and U) = 60/200 = 0.3, P(M) P(U) = = 0.3, so they are independent.

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

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. 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. 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) 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? 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 Non-mutually Exclusive Events P(A and B) ≠ 0. P(A and B) ≠ 0. P(A or B) = P(A) + P(B) - P(A and B). P(A or B) = P(A) + P(B) - P(A and B).

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 B Venn diagram for events A and B. The rectangle represents all possible outcomes of an experiment A0.6 A0.4 Venn diagram for P(A) = 0.4.

Going Beyond the Contingency Table (cont.) A B A B A B 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 Venn diagram of mutually exclusive events. P(A and B) = 0 and P(A or B) = P(A) + P(B).

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 Special Additive Rule If A and B are mutually exclusive then P(A or B) = P(A) + P(B) 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). 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 – 60/200 P(M or U) = P(M) + P(U) – P(M and U) = 120/ /200 – 60/200 = 0.80 = 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)? 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) = – 0.1 = 0.6 Example 4.3.3: In Datacomp Survey, find P(M or F). 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 – 0/200 = – 0 = = P(M) + P(F) = 1.0

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

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.

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)

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) =

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.

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: B B B E1E1 E2E2 EnEn Example 4.4.1: Draw a tree diagram for the Datacomp Survey data.

Tree Diagrams (cont.) Rules for Tree Diagram 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 # 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 i th path is: Rule # 2:The posterior probability for the i th path is:

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 Defective Shift 1 2 Defective 3 (.06) (.08) (.15) (.5) (.2) (.3)

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