Chapter 4&5_Tutorial_W24_Solution

Department of Mathematics and Statistics STAT2910-01: Statistics for Sciences Faculty of Science University of Windsor Tutorial for Chapter 4 and 5 Questions for Chapter 4 Exercise 1. Of the delegates at a convention, 60% attended the breakfast forum, 70%attended the dinner speech, and 40% attended both events. Define events A and B as follows: A: attended the breakfast forum B: attended the dinner speech What is the probability that a randomly selected delegate either attended the breakfast forum, or attended the dinner speech, or attended both? Solution 1. The requested probability is given by P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) 0 . 60 + 0 . 70 − . 040 = 0 . 90 . Exercise 2. Heidi prepares for an exam by studying a list of 15 problems. She can solve 9 of them. For the exam, the instructor selects 7 questions at random from the list of 15. What is the probability that Heidi can solve all 7 problems on the exam? Solution 2. The instructor can select 7 out of the lsit of 15 questions in ( 15 7 ) ways. If Heidi is able to answer all 17 questions, the instructor must choose 7 questions out of the 9 that Heidi can answer, and none of the 6 questions that Heidi cannot answer. The number of ways in which this event can occur is ( 9 7 )( 6 0 ) . Hence, the probability that Heidi can answer all 17 questions is ( 9 7 )( 6 0 ) ( 15 7 ) = 36 6435 Exercise 3. a. How many different combinations of 5 students can be drawn from a class of 25 students? b. How many permutations of 3 colours can be drawn from a group of 20 colours? 1

Solution 3. a. There are ( 25 5 ) = 35130 combinations. b. There are P 3 20 = 6840 permutations. Exercise 4. A psychologist tests Grade 7 students on basic word association skills and number pattern recognition skills. Let W be the event a student does well on the word association test. Let N be the event a student does well on the number pattern recognition test. A student is selected at random, and the following probabilities are given: P ( W ∩ N ) = 0 . 25 , P W ∩ N ∁ = 0 . 15 , P W ∁ ∩ N = 0 . 10 , P W ∁ ∩ N ∁ = 0 . 50 a. What is the probability that the randomly selected student does well on the word association test? b. What is the probability that the randomly selected student does well on the number pattern recognition test? c. What is the probability that the randomly selected student does well on at least one of the tests? d. If the randomly selected student does well on the word association test, what is the probability he or she will also do well on the number pattern recognition test? e. If the randomly selected student does well on the number pattern recognition test, what is the proba- bility he or she will also do well on the word association test? f. Are the events W and N mutually exclusive? Justify your answer. g. Are the events W and N independent? Explain. Solution 4. a. The desired probability is given by P ( W ) = P ( W ∩ N ) + P W ∩ N ∁ = 0 . 25 + 0 . 15 = 0 . 40 b. The desired probability is given by P ( N ) = P ( W ∩ N ) + P W ∁ ∩ N = 0 . 25 + 0 . 10 = 0 . 35 c. The desired probability is given by P ( W ∪ N ) = P ( W ) + P ( N ) − P ( W ∩ N ) = 0 . 40 + 0 . 35 + − 0 . 25 = 0 . 5 Or P ( W ∪ N ) = 1 − P W ∁ ∩ N ∁ = 1 − 0 . 5 = 0 . 50 2

d. The desired probability is given by P ( N | W ) = P ( W ∩ N ) P ( W ) = 0 . 25 0 . 40 = 0 . 625 e. The desired probability is given by P ( W | N ) = P ( W ∩ N ) P ( N ) = 0 . 25 0 . 35 = 0 . 7143 f. No, they are not mutually exclusive because P ( W ∩ N ) ̸ = 0. g. No, they are not independent. For example P ( W | N ) = 0 . 7143 ̸ = P ( W ). Exercise 5. A researcher studied the relationship between the salary of a working woman with school-aged children and the number of children she had. The results are shown in the following probability table: Salary 2 or fewer Children More than 2 Children High salary 0.13 0.2 Medium salary 0.20 0.10 Low salary 0.30 0.25 Let A denote the event that a working woman has two or fewer children, and let B denote the event that a working woman has a low salary. a. What is the probability that a working woman has two or fewer children? b. What is the probability that a working woman has a low salary? c. What is the probability that a working woman has two or fewer children and has a low salary? d. What is the probability that a working woman either has two or fewer children or has a low salary? e. If a working woman has two or fewer children, what is the probability that she has a low salary? f. If a working woman has a low salary, what is the probability that she has two or fewer children? g. From this information, can one conclude that the salary of a working woman with school-aged children and the number of children she has are independent events? Explain. 3

Solution 5. a. The requested probability is P ( A ) = 0 . 13 + 0 . 20 + 0 . 30 = 0 . 63 b. The requested probability is P ( B ) = 0 . 30 + 0 . 25 = 0 . 55 c. The requested probability is P ( A ∩ B ) = 0 . 30 d. The requested probability is P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) = 0 . 63 + 0 . 55 − 0 . 30 = 0 . 88 e. The requested probability is P ( B | A ) = P ( A ∩ B ) P ( A ) = 0 . 30 0 . 63 = 0 . 4762 f. This probability is given by P ( A | B ) = P ( A ∩ B ) P ( B ) = 0 . 30 0 . 55 = 0 . 5455 g. No. For example P ( A | B ) = 0 . 5455 ̸ = P ( A ) = 0 . 63 Exercise 6. A federal agency is trying to decide which of two waste management projects to investigate as the source of air pollution. In the past, projects of the first type were in violation of air quality standards with probability 0.3 on any given day, while projects of the second type were in violation of air quality standards with probability 0.25 on any given day. It is not possible for both projects to pollute the air in one day. Let A i , i = 1 , 2, denote that project of type i was in violation of air quality standards. a. Find the probability of an air pollution problem being caused by either the first project or the second project. b. If the first project is violating air quality standards, what is the probability the second project is also violating federal air quality standards? 4

Solution 6. a. Since A 1 and A 2 are mutually exclusive, the requested probability is given by P ( A 1 ∪ A 2 ) = P ( A 1 ) + P ( A 2 ) = 0 . 30 + 0 . 25 = 0 . 55 b. Since A 1 and A 2 are mutually exclusive, the requested probability is given by P ( A 1 | A 2 ) = P ( A 1 ∩ A 2 ) P ( A 2 ) = 0 0 . 30 = 0 Exercise 7. An experiment was conducted in which rats could choose to enter one of two corridors, A or B. A random sample of three rats is selected. Let X = number of rats that select corridor B. a. Assuming the rats select their favourite corridor independently of one another and that the two corri- dors are equally likely to be selected, find the probability distribution of X . b. What is the probability that, at most, one rat selects corridor B? c. What is the probability that at least one rat selects corridor B? Solution 7. a. The probability distribution of X is given by p (0) = P ( A ∩ A ∩ A ) = P ( A ) P ( A ) P ( A ) = 1 2 3 = 0 . 125 p (1) = P ( A ∩ A ∩ B ) + P ( A ∩ B ∩ A ) + P ( B ∩ A ∩ A ) = 3 1 2 3 = 0 . 375 p (2) = P ( B ∩ B ∩ A ) + P ( B ∩ A ∩ B ) + P ( A ∩ B ∩ B ) = 3 1 2 3 = 0 . 375 p (3) = P ( B ∩ B ∩ B ) = 0 . 125 As a table x 0 1 2 3 p ( x ) 0.125 0.375 0.375 0.125 Or graphically 5