Unformatted text preview: Chapter 5 Probability Concepts 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 5.1.1: Examples of experiments 2 Flipping a coin Rolling two dice Taking an exam Observing the number of arrivals at a driveup window over a 5minute Observing period period Events and Probability (cont.) Sample Space The list of all possible outcomes of an experiment is called the sample space. Example 5.1.2: Example of sample space Flipping a coin twice results in one of four possible outcomes. These possible outcomes are HH, HT, TH, TT. Therefore, sample space = {HH, HT, TH, TT}. If there are n outcomes of an experiment, sample space lists all n outcomes. 3 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 5.1.3: Examples of experiments and some corresponding Example events events Experiment: Rolling two dice; events: A = rolling a total of 7, B = rolling a events: total greater than 8, C = rolling two 4s. Experiment: Taking an exam; events: A = pass, B = fail. events: Experiment: Observing the number of arrivals at a driveup window over a 5minute period; events: A0 = no arrivals, A1 = seven arrivals, etc. events: 4 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. 5 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: m P ( A) = n This assumes all n possible outcomes have an equal chance of occurring. Example 5.1.4: Toss a nickel and a dime. The sample space (i.e., the list of Example 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. probability, 6 Events and Probability (cont.) Relative Frequency Approach Observe an experiment n times and count the number of times event A Observe occurs, m. occurs, P ( A) = m n The relative frequency definition of a probability is based on past observations Example: Your last 200 customers 7 140 females 60 males P(next customer is a female) is 140 / 200 = .7 Based on these observations, there is a 70% chance that the next customer is a female Relative Frequency Definition Example 5.1.5: A production process has been in operation in for 250 days and has been accidentfree for 220 days. If event A is a randomly chosen accidentfree 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 5.1.6: Example of subjective probability: The probability that it will rain Example today is 50%.
8 Basic Concepts Contingency Table (also called CrossTab Table) Contingency tables are used to record and analyze the relationship between two variables. Example 5.2.1: Datacomp Survey: Datacomp recently conducted a survey of 200 selected purchasers of their newly introduced laptop computer to obtain a genderandage profile of its new customers. The data are summarized in the following contingency table.
Age (Years) < 30 Gender
9 30 45 (B) 20 30 50 > 45 (O) 40 10 Total 120 80 (U) 60 40 Male (M) Female (F) Some Events: Total 100 M = a male is selected F = a female is selected U = the person selected is under 30 50 200 B = the person selected is between 30 & 45 O = the person selected is over 45 Basic Concepts (cont.) Marginal Probability Marginal probability is the probability of one event, regardless of the other events. Example 5.2.2: 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 10 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. This For example, A = it rains tomorrow, A = it does not rain tomorrow. For Example 5.2.3: 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) 11 Basic Concepts (cont.) Joint Probability The probability of the occurrence of two events at the same time Example 5.2.4: In Datacomp Survey, what proportions are males Example between 30 and 45? That is, find the probability of selecting a person who is a male and between 30 and 45. who P(M and B) = 20/200 = 0.10 12 Example 5.2.5: The probability of selecting a person who is a female Example and under 30 is P(F and U) = 40/200 = 0.20. P(F Basic Concepts (cont.) Either of Two Events The probability of either event A or event B occurring is written as P(A The or B). or Example 5.2.6: In Datacomp Survey, the probability of selecting a Example person who is male or under 30 is P(M or U) = (120+10060) / 200 = 0.80. 0.80 Conditional Probability Whenever you are given information and are asked to find a probability Whenever based on this information, the result is a conditional probability. based This probability is written as P(AB) and read as “probability of A given This B”. B”. Example 5.2.7: In Datacomp Survey, what is the probability that a Example randomly selected customer is male given that he is under 30? randomly 13 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 (assuming P(B  A) = P(B) (assuming P(A) ≠ 0), or (assuming P(A and B) = P(A) • P(B) Example 5.2.8: In Datacomp Survey, are events M and U independent? 14 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 P(M independent. independent. Basic Concepts (cont.) Dependent Events Events that are not independent are dependent events. Events dependent 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) (A (B 15 Basic Concepts (cont.) Mutually Exclusive Events If the two events cannot occur at the same time they are said to be If mutually exclusive. mutually Events A and B are mutually exclusive if their joint probability is zero, Events that is, P(A and B) = 0. that NonMutually Exclusive Events P(A and B) ≠ 0 16 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) If Example 5.3.1: In Datacomp Survey, what is the probability of selecting Example a person who is male or under 30? That is, find P(M or U). P(M P(M or U) = P(M) + P(U) – P(M and U) = 120/200 + 100/200 – 60/200 60/200 = 0.80 0.80 17 Example 5.3.2: The probability of event A is 0.5 and the probability of Example event B is 0.2. If P(A and B) is 0.1, what is P(A or B)? event P(A or B) = P(A) + P(B) – P(A and B) = 0.5 + 0.2 – 0.1 = 0.6 Example 5.3.3: In Datacomp Survey, find P(M or F). P(M or F) = P(M) + P(F) – P(M and F) = 120/200 + 80/200 – P(M Probability Rules (cont.) General Multiplicative Rule P ( A and B ) = P ( A  B ) ⋅ P ( B ) = P ( B  A) ⋅ P ( A) Special Multiplicative Rule If events A and B are independent then: P ( A and B ) = P ( A) ⋅ P ( B ) Example 5.3.6: Let P(A) = 0.6, P(B) = 0.2, and P(AB) = 0.1. Find P(A and B) P ( A and B ) = P ( A  B ) ⋅ P ( B ) = 0.1 ⋅ 0.2 = 0.02
18 Rules (cont.) General Conditional Probability Rule P ( A and B) P( B) P ( A and B) P( B  A) = P( A) P( A  B) = [ assuming P(B) ≠ 0], and [ assuming P(A) ≠ 0] Special Conditional Probability Rule If events A and B are independent then:
19 P ( A  B) = P( A), and P ( B  A) = P( B ) Examples Example 5.3.4: If P(A and B) = 0.4 and P(B) = 0.8, find P(AB). P ( A and B ) 0.4 P( A  B) = = = 0 .5 P( B) 0 .8 Example 5.3.5: 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. 20 P ( A and B ) 0.2 P( A  B) = = = 0.5 ≠ P( A) P( B) 0 .4 P ( A and B ) 0.2 P ( B  A) = = = 0.67 ≠ P ( B ) P ( A) 0 .3
Events A and B are not independent. Components in Series The system A B C There is a 2% chance of failure for each of the three components 21 The system fails if one or more components fail Components in Series P(system fails) = 1 – P(system doesn’t fail) This is 1 – P(A works and B works and C works) 1 – [P(A works) · P(B works) · P(C works)] 1 [(.98) ·(.98)·(.98)] = .059 (roughly 6%) 22 Components in Parallel The system
A B C There is a 2% chance of failure for each of the three components 23 The system fails if all three components fail Components in Parallel P(system fails) = P(A fails and B fails and C fails) This is P(A fails) · P(B fails) · P(C fails) = (.02)(.02)(.02) = .000008 (8 in a million chance) This is called builtin redundancy
24 Going Beyond the Contingency Table 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 5.3.7: 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)]? 25 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(BA) = 3/51. P(A and B) = (4/52)(3/51) = 0.0045. 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 5.3.8: Let A = selecting a king on the first draw, and B = selecting a king on the second draw. What is P(BA)? There are still 52 cards in the deck. So, P(BA) = 4/52. Using Excel to Construct a Contingency Table KPK Data Analysis > Qualitative Data Charts > Contingency Table.
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This note was uploaded on 04/15/2011 for the course DSCI 2710 taught by Professor Hossain during the Spring '08 term at North Texas.
 Spring '08
 Hossain

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