Probability Theory And Statistics The Course of Lectures

Probability Theory And Statistics The Course of Lectures. 2/8/2018 1

Seminar 4. Bayes’ Theorem. Random Variables. Probability Distributions for Discrete Random Variables. 2/8/2018 2

Question n There are 6 urn with balls of different colors n n a) b) Three urns contain 5 white and 5 black balls Two urns contain 3 white and 7 black balls One urn contains 10 black balls Someone chooses by chance one of the urns and takes a ball from the urn. Find the probability of event “the ball is white” The ball appears to be white. Find the probability of event “the ball was taken from an urn of the first type” 2/8/2018 3

Question n Three bank clerks develop the bills. n n n n Iren develops 50% of all bills, Lise 30% and Masha 20%. Iren makes mistake in 2% of bills she developed, Lise – in 3% and Masha – in 10% The incorrectly developed bill was found. Calculate probability that this bill was developed by Lise. 2/8/2018 4

Question n In a large city, 8% of the inhabitants have contracted a particular disease. A test for this disease is positive in 80% of people who have the disease and is negative in 80% of people who do not have the disease. What is the probability that a person for whom the test result is positive has the disease? 2/8/2018 5

Question n n A campus student club distributed material about membership to new students attending an orientation meeting. Of those receiving this material, n n n Subsequently, it was found n n 40% were men and 60% were women. the 7% of the men and 9% of the women who received this material joined the club. Find the probability that a randomly chosen new student who receives the membership material will join the club. Find the probability that a randomly chosen new student who joins the club after receiving the membership material is a woman 2/8/2018 6

Question n n A professor finds that she awards a final grade of A to 20% of the students. Of those who obtain a final grade of A, 70% obtained an A in the midterm examination. Also, 10% of students who failed to obtain a final grade of A earned an A in the midterm exam. What is the probability that a student with an A on the midterm examination will obtain a final grade of A? 2/8/2018 7

Question n n A restaurant manager classifies customers as well dressed, moderately dressed, or poorly dressed, and finds 50%, 40%, and 10% respectively of all customers fall into these categories. The manager found that wine was ordered n n n by 70% of the well dressed, by 50% of the moderately dressed, and by 30% of the poorly dressed customers What is the probability that a randomly chosen customer orders wine? If wine is ordered, what is the probability that the person ordering was well dressed? If wine is ordered, what is the probability that the person ordering was not well dressed? 2/8/2018 8

Question n n A record store owner assesses customers entering the store as high school age, college age, or older, and finds 30%, 50%, and 20% respectively of all customers fall into these categories. The owner also found that purchases were made by 20% of high school age customers, by 60% of college age customers, and by 80% of older customers. What is the probability that a randomly chosen customer entering the store will make a purchase? If a randomly chosen customer makes a purchase, what is the probability that this customer is high school age? If a randomly chosen customer makes a purchase, what is the probability that this customer is not high school age? 2/8/2018 9

Question n n The grades of a freshman college class, obtained at the end of their first year of college, were analyzed. 70% of the students in the top quarter of the college class had graduated in the upper 10% of their high school class, as had 50% of the students in the middle half of the college class and 20% of the students in the bottom quarter of the college class. What is the probability that a randomly chosen freshman graduated in the upper 10% of his or her high school class? What is the probability that a randomly chosen freshman who graduated in the upper 10% of his or her high school class will be in the top quarter of the college class? What is the probability that a randomly chosen freshman who did not graduate in the upper 10% of his or her high school class will not be in the top quarter of the college class? 2/8/2018 10

Question n State, with reasons, whether each of the following statements is true or false: a) The complement of the union of two events is the intersection of their complements. b) The sum of the probabilities of collectively exhaustive events must equal 1. c) The number of combinations of x objects chosen from n is equal to the number of combinations of (n — x) objects chosen from n, where 2/8/2018 11

Question n State, with reasons, whether each of the following statements is true or false: d) If A and B are two events, the probability of A given B is the same as the probability of B given A if the probability of A is the same as the probability of B. e) If an event and its complement are equally likely to f) g) 2/8/2018 occur, the probability of that event must. 5. If A and B are independent, then A and B must be independent. If A and B are mutually exclusive, then A and B must be mutually exclusive 12

Question n State, with reasons, whether each of the following statements is true or false: a) The probability of the union of two events cannot be less than the probability of their intersection. b) The probability of the union of two events cannot be more than the sum of their individual probabilities. c) The probability of the intersection of two events cannot be greater than either of their individual probabilities An event and its complement are mutually exclusive. 2/8/2018 13

Question n State, with reasons, whether each of the following statements is true or false: d) The individual probabilities of a pair of e) f) 2/8/2018 events cannot sum to more than 1. If a pair of events are mutually exclusive, they must also be collectively exhaustive. If a pair of events are collectively exhaustive, they must also be mutually exclusive. 14

Question An insurance company estimated that 30% of all automobile accidents were partly caused by weather conditions, and that 20% of all automobile accidents involved bodily injury. n Further, of those accidents that involved bodily injury, 40% were partly caused by weather conditions. n What is the probability that a randomly chosen accident both was partly caused by weather conditions and involved bodily injury? n Are the events "partly caused by weather conditions" and "involved bodily injury" independent? n If a randomly chosen accident was partly caused by weather conditions, what is the probability that it involved bodily injury? n What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury? 2/8/2018 15 n

Question n n Based on a survey of students on a large campus, it was estimated that 35% of the students drink at least once a week in campus bars, and that 40% of all students have grade point averages of B or better. Further, of those who drink at least once a week in campus bars, 30% have a B average or better. What is the probability that a randomly chosen student both drinks at least once a week in campus bars and has a B average or better? What is the probability that a randomly chosen student, who has a B average or better, drinks at least once a week in campus bars? What is the probability that a randomly chosen student has at least one of the charac teristics "drinks at least once a week in campus bars" or "B average or better"? 2/8/2018 16

Thank you for your attention! 2/8/2018 17