Giải tích 1 - Chapter 4: Introduction to probability

Chapter 4 INTRODUCTION TO PROBABILITY Nguyen Tien Dung, MBA School of Economics and Management Website: https://sites.google.com/site/nguyentiendungbkhn Email: dung.nguyentien3@hust.edu.vn Main Contents 4.1 EXPERIMENTS, COUNTING RULES AND ASSIGNING PROBABILITIES 4.2 EVENTS AND THEIR PROBABILITIES 4.3 SOME BASIC RELATIONSHIPS OF PROBABILITY 4.4 CONDITIONAL PROBABILITY 4.5 BAYES’ THEOREM © Nguyễn Tiến Dũng Applied Statistics in Business 2 4.1 EXPERIMENTS, COUNTING RULES, AND ASSIGNING PROBABILITIES ● Sample space for an experiment: ● the set of all experiment outcomes ● An experiment outcome is also called a sample point © Nguyễn Tiến Dũng Applied Statistics in Business 3 Probability As A Numerical Measure Of The Likelihood Of An Event Occurring ●S = {Head; Tail} ●S = {Defective, Nondefective} ●S = {1, 2, 3, 4, 5, 6} © Nguyễn Tiến Dũng Applied Statistics in Business 4 Counting Rules, Combinations, and Permutation ● Counting Rules ● Being able to identify and count the experimental outcomes is a necessary step in assigning probabilities. ● Multiple-step experiments ● Example 1: Toss 2 coints ● Example 2: Kentucky Power & Light Company Construction Project (KP&L Project) ● Tree diagram ● a graphical representation that helps in visualizing a multiple-step experiment. © Nguyễn Tiến Dũng Applied Statistics in Business 5 The Tree Diagram For the Tossing-2-Coint Experiment © Nguyễn Tiến Dũng Applied Statistics in Business 6 The Tree Diagram for the KP&L Project © Nguyễn Tiến Dũng Applied Statistics in Business 7 Combinations ● Example: There are 4 football teams, playing in a tournament. Each will meet the rest once. How many football matches will the tournament have? ●Do it in 2 ways: manually and using C(N,n). © Nguyễn Tiến Dũng Applied Statistics in Business 8 Permutations ●Example: There are 5 digits: 1, 2, 3, 4, 5. How many two-digit numbers could be formed from these five digits. © Nguyễn Tiến Dũng Applied Statistics in Business 9 Assigning Probabilities © Nguyễn Tiến Dũng Applied Statistics in Business 10 Assigning Probabilities ●Classical method ● No. of favourable outcomes (computation) / Total possible outcomes ● When all experimental outcomes are equally likely ●Relative frequency ● Actual data ● Frequency of occurrences of interested outcomes / Total actual outcomes ●Subjective method ● A method of assigning probabilities on the basis of judgment © Nguyễn Tiến Dũng Applied Statistics in Business 11 Classical method ●Appropriate when all the experimental outcomes are equally likely. ● If n experimental outcomes are possible, a probability of 1/n is assigned to each experimental outcome. ●Example 1: toss a coin ●Example 2: roll a dice © Nguyễn Tiến Dũng Applied Statistics in Business 12 Relative frequency method ● Appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times. ● Example: a study of waiting times in the X-ray department for a local hospital. A clerk recorded the number of patients waiting for service at 9:00 a.m. on 20 successive days and obtained the following results. © Nguyễn Tiến Dũng Applied Statistics in Business 13 Number of Waiting Number of Days Outcome Occurred 0 2 1 5 2 6 3 4 4 3 Total 20 Number of Waiting Probability of Outcome Occurred 0 2/20 = 0.1 1 5/20 = 0.25 2 6/20 = 0.3 3 4/20 = 0.2 4 3/20 = 0.15 Total 20/20 =1.00 Assigning Probabilities by the Relative Frequency Method © Nguyễn Tiến Dũng Applied Statistics in Business 14 4.2 EVENTS AND THEIR PROBABILITIES ● An event is a collection of sample points ● KP&L Project ● C = {(2,6), (2, 7), (2,8), (3,6), (3,7), (4,6)} ● Event C includes many sample points ● One event may be comprised of many events ● L = the event that the projects is completed in LESS than 10 months ● L = {(2,6), (2,7), (3, 6)} ● M = the event that the project is completed in 10 months or more than 10 months ● M = {(2,8), (3,7), (4,6) ● C = {L, M} ● Probability of an event ● The sum of the probability of the sample points in the event. © Nguyễn Tiến Dũng Applied Statistics in Business 15 4.3 SOME BASIC RELATIONSHIPS OF PROBABILITY ●Complement of an event ●P(A) = 1 – P(Ac) © Nguyễn Tiến Dũng Applied Statistics in Business 16 Union of Two Events ●The union of A and B is the event containing all sample points belonging to A or B or both. The union is denoted by A  B. © Nguyễn Tiến Dũng Applied Statistics in Business 17 Intersection of Two Events ●Given two events A and B, the intersection of A and B is the event containing the sample points belonging to both A and B. The intersection is denoted by A B. © Nguyễn Tiến Dũng Applied Statistics in Business 18 Addition Law ● 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵) ● Example: a small assembly plant with 50 employees. Each worker is expected to complete work assignments on time and in such a way that the assembled product will pass a final inspection. or assembling a defective product. ● 5 of the 50 workers completed work late, ● 6 of the 50 workers assembled a defective product, ● 2 of the 50 workers both completed work late and assembled a defective product. ● Questions: Take randomly an employee. What is the probability taking an employee with a poor performance rating? © Nguyễn Tiến Dũng Applied Statistics in Business 19 Addition Law for Mutually Exclusive Events ●Two events are said to be mutually exclusive if the events have no sample points in common. P(A  B) = P(A) + P(B) © Nguyễn Tiến Dũng Applied Statistics in Business 20 4.4 CONDITIONAL PROBABILITY ●The probability of event A given the condition that event B has occurred: P(A | B) ●Example: ● There are 5 white balls and 3 black balls in a box. Draw randomly 2 balls consecutively. What is the probability of having 2 white balls? © Nguyễn Tiến Dũng Applied Statistics in Business 21 ●Example: The situation of promotion status of male and female officers in a police force © Nguyễn Tiến Dũng Applied Statistics in Business 22 Joint Probability Table ●M = event an officer is a man ● A = event an officer is promoted ● P(M  A)= 288/1200 = 0.24 © Nguyễn Tiến Dũng Applied Statistics in Business 23 © Nguyễn Tiến Dũng Applied Statistics in Business 24 288 288 / 1200 ( ) 0.24 ( | ) 0.30 960 960 / 1200 ( ) 0.80 P A M P A M P M       36 36 / 1200 ( ) 0.03 ( | ) 0.15 240 240 / 1200 ( ) 0.2 P A W P A W P W      Conditional Probability © Nguyễn Tiến Dũng Applied Statistics in Business 25 ( ) ( | ) ( ) P A B P A B P B   ( ) ( | ) ( ) P A B P B A P A   Independent Events ●Two events A and B are independent if P(A | B) = P(A) or P(B | A) = P(B). ●Example: ● Toss 2 coins: A = head of the first coin; B = head of the second coin © Nguyễn Tiến Dũng Applied Statistics in Business 26 Multiplication Law ● In general ● P(A  B) = P(A).P(B | A) = P(B).P(A | B) ● For independent events ● P(A  B) = P(A).P(B) ● Example: A service station manager who knows from past experience that 80% of the customers use a credit card when they purchase gasoline. What is the probability that the next two customers purchasing gasoline will each use a credit card? © Nguyễn Tiến Dũng Applied Statistics in Business 27 ●C1 = the event the first customer use the credit card ●C2 = the event the second customer use the credit card ●C1 and C2 are independent events ●P(C1.C2) = P(C1).P(C2) = (0.8).(0.8) = 0.64 © Nguyễn Tiến Dũng Applied Statistics in Business 28 4.5 BAYES’ THEOREM ●Meaning: © Nguyễn Tiến Dũng Applied Statistics in Business 29 Bayes’ Theorem © Nguyễn Tiến Dũng Applied Statistics in Business 30 ● Example: ● Two suppliers provide products to a business customer ● Supplier 1 provides 65% of product quantity ● Supplier 2 provides 35% of product quantity ● 98% products of Supplier 1 are met quality standards (Good), 2% are Bad. ● 95% products of Supplier 2 are met quality standards (Good), 5% are Bad. ● Take randomly a product provided and check its quality. The result is that product failed to meet quality standards. ● What is the probability that the product was come from Supplier 1? Supplier 2? © Nguyễn Tiến Dũng Applied Statistics in Business 31 Probability Tree for Two-supplier Example © Nguyễn Tiến Dũng Applied Statistics in Business 32 Tabular Approach © Nguyễn Tiến Dũng Applied Statistics in Business 33 Exercises for Homework ●4.1: Exercises 6, 7, 9, 10, 13 ●4.2: Exercises 16, 17, 19, 21 ●4.3: Exercises 24, 25, 28 ●4.4: Exercises 32, 33, 37 ●4.5: Exercises 40, 41, 42 © Nguyễn Tiến Dũng Applied Statistics in Business 34