Introduction to Business Statistics I (QM 120) Homework # 4

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1 Introduction to Business Statistics I (QM 120) Homework # 4 Problem # 1 The sample space for an experiment contains five simple events with probabilities is shown in the table. Find the probability of each of the following events: Simple events Probability a. Either 1, 2, or 3 occurs P(1 or 2 or 3) = P(1) + P(2) + P(3) = =.55 b. Either 1, 3, or 5 occurs P(1 or 3 or 5) = P(1) + P(3) + P(5) = =.50 c. 4 does not occur P(4 doesn t occur) = 1- P(4) = =.70 Problem # 2 Compute each of the following: a. P 4, P 2, P 0, P3 and P 1 n n! Pr =, I will do the first one only ( n r )! 9 9! P 4 = = = = 3024 (9 4)! P = 42, P = 1, P = 6, P = b. C 4, C 2, C 0, C3 and C 1 n n! Cr =, again I will do the first one only r!( n r )! 9 9! C 4 = = = = 126 4!(9 4)! C = 21, C = 1, C = 1, C =

2 Problem # 3 Two fair dice are tossed, and the face on each die is observed. a. List the sample space of this experiment. 1 st die 2 nd die 1 st die 2 nd die 1 st die 2 nd die 1 st die 2 nd die 1 st die 2 nd die 1 st die 2 nd die b. Find the probability of each of the following events: Each event has a probability of 1/36 A: {3 showing up} There are 11 outcomes that number 3 shows up, so P(A) = 11/36 P(B) = 1/6 B: {sum of two numbers showing is 7} P(B) = 1/2 Problem # 4 C: {sum of two numbers showing is even} Two marbles are drawn at random without replacement from a box containing two blue marbles and three red marbles. a. List the sample space for this experiment. S = { blue blue, blue red, red blue, red red} b. Determine the probability of observing each of the following events: Let B1 : The first ball drawn be blue R1 : The first ball drawn be red B2: the second ball drawn be blue R2: the second ball drawn be red A: {2 blue marbles are drawn}. P(B1 B2) = P(B1).P(B2 B1) = 2/5. 1/4 = 1/10

3 B: {a red and blue marble are drawn}. P(B R) = P(B1).P(R2 B1) + P(R1).P(B2 R1) = 2/5. 3/4 + 3/5. 2/4 = 6/10 C: {2 red marbles are drawn}. P(R1 R2) = P(R1).P(R2 R1) = 3/5. 2/4 = 3/10 Note that we were able to find the last probability by using the compliment event technique ( 1 [1/10 + 6/10] = 3/10) or just to make sure that our solution is correct we can check that the probabilities of all the three events add up to 1 Problem # 5 Simulate the experiment described in problem 4 using any five identically shaped objects, two of which are blue and three red. Mix the objects, draw two, record the results, and then replace the objects. Repeat the experiment a large number of times (at least 100). Calculate the proportion of times events A, B, and C occur. How do these proportions compare with the probabilities you calculated in problem 4. You should be able to do this problem by your own it is really fun! Problem # 6 A sample of 1000 families showed that 34 of them own no cars, 208 own one car each, 376 own two cars each, 265 own three cars each, and 117 own four or more cars each. Write the frequency distribution table for this problem. Calculate the relative frequencies for all categories. Suppose one family is randomly selected from these 1000 families. Find the probability that this family owns a- Two cars b- Three or more cars c- Two or fewer cars We have the following probabilities: P(0 car) = 34/1000, P(1 car) = 208/1000, P(2 cars) = 376/1000 P(3 cars) = 265/1000, P(> 4 cars) = 117/1000 a- P(2 cars) =.376 b- P(> 3 cars) = P(3 cars) + P(>4 cars) = =.382 c- P(2 or fewer cars) = P(0 car) + P(1 car) + P(2 cars) = =.618

4 Problem # 7 Five hundred employees were selected from two small banks that are proposing a merger, and they were asked whether they are in favor or against the merger. Based on this information, the following two way classification table was prepared. In favor Against Total Bank A Bank B Total a If one employee is selected at random from these 500 employees, find the probability that i. the employee is from bank A P(Bank A) = 300/500 =.6 ii. the employee is in favor P(In favor) = 375/500 =.75 iii. the employee is in favor given the employee is from bank A P(In favor Bank A) = 225/300 =.75 iv. the employee is from bank B given that he/she is against the proposal P(Bank B Against) = 50/125 =.4 v. the employee is from bank B and is in favor P(In favor I Bank B) = 150/500 =.3 vi. of the union of events ʺbank Bʺ and ʺin favorʺ P(In favor U Bank B) = P(In favor) + P(Bank B) P(In favor I Bank B) = =.85 b Are the events ʺBank Aʺ and ʺIn favorʺ mutually exclusive? What about the events ʺIn favorʺ and ʺAgainstʺ? Why or why not? P(In favor I Bank A) doesn t equal 0, so they are not mutually exclusive. P(In favor I Against) equals 0, so they are mutually exclusive. c Are the events ʺBank Bʺ and ʺIn favorʺ independent? Why or why not? The two events are independent if P(Bank B) = P(Bank B In favor). P(Bank B) = 200/500 =.4, P(Bank B In favor) = 150/375 =.4 So, they are independent.

5 Problem # 8 An experiment can result in one of the five equally likely simple events, E1, E2,, E5. Events A, B, and C are defined as follows: A: E1, E3 B: E1, E2, E3, E4. C: E4,E5 Find the probabilities associated with the following compound events a c A b A I B c B I C d A U B e B C f A B c g ( A I B) h Are events A and B independent? i Are events A and B mutually exclusive? a P(A c ) = 1 P(A) = 1 2/5 = 3/5 A b P(A I B) = P(A) = 2/5 c P(B I C) = 1/5 d P(A U B) = P(A) + P(B) P(A I B) = P(B) = 4/5 e P(B C) = P(BI C)/P(C) = 1/5 = 1 2/5 2 f - P(A B) = P(A I B)/P(B) = 2/5 2 B = 3/5 3 g P([A I B] c ) = 1 P(A I B) = 1 2/5 = 3/5 h Dependent, because P(A) = 2/5 doesn t equal P(A B) = 2/3 I No, because P(A I B) doesn t equal to 0 Problem # 9 E 2 E 1 E 3 E 4 E 5 C A smoke detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is.96; by device B.97; and by both devices,.94. a If smoke is present, find the probability that the smoke will be detected by device A or device B or both. b Find the probability that the smoke will not be detected. Think of this problem as an electrical circuit with two light bulbs connected in parallel as shown in the figure below

6 bulb A can be on with P=.96 bulb B can be on with P=.97 both bulbs can be on wit P =.94 a. In this case the probability that the smoke will be detected = the probability at least one of the bulbs will be on; which is P(A U B) = P(A) + P(B) P(A I B) = =.99 b. The smoke will not be detected = none of the lights will be on (Dark) and that is 1 - P(A U B) = =.01 Problem # 10 An experiment consists of tossing a single die and observing the number of dots that shows on the upper face. Events A, B, and C are defined as follows: A: Observe a number less than 4 B: Observe a number less than or equal to 2 C: Observe a number greater than 3 Find the probabilities associated with these compound events a S b A B c B d A I B I C e AI B f B U C g B I C h AU C i Are events ʺAʺ and ʺBʺ independent? Mutually exclusive? j Are events ʺAʺ and ʺCʺ independent? Mutually exclusive? Let us first find the sample space of this experiment. S = {1, 2, 3, 4, 5, 6} The events are:- A = {1, 2, 3}, B = {1, 2}, c = {4, 5, 6} a. P(S) = 1 b. P(A B) = P(A I B)/P(B) = 1 c. P(B) = 2/6 d. P(A I B I C) = 0 e. P(A I B) = 2/6 f. P(B U C) = 5/6 g. P(B I C) = 0 h. P(A U C) = 1 i. P(A) = 3/6, P(A B) = 1, so they are dependent. Since the P(AI B) 0, they are not mutually exclusive j. P(A) = 3/6, P(A C) = 0, so they are dependent. Since the P(AI B) = 0, we conclude that they are mutually exclusive

7 Problem # 11 A journalist submitted two different articles to a newspaper. He knows from his past experience that the newspaper will publish the first one with a probability of 0.7, and the probability of getting the second one published is 0.4. Assuming the Editor in Chief will publish the articles independently, what is the probability that: a. both articles will be published b. at least one article will be published c. none of the articles will be published d. The news paper will publish the second article only Let A = the first article will be published B = the second article will be published a. S = { AB, AB, AB, AB} with the following probabilities {.7x.4,.3x.4,.7x.6,.3x.6} respectively. So the probability needed in this part is P(AB) and it is P(AB) = P(A)P(B) =.7x.4 =.28 b. P(A U B) = P(A) + P(B) P(AB) = =.82 Or, it is simply P(AB) + P(AB) + (AB) = =.82 c. Again, it is either 1- P(A U B) =.18 or, P(AB) =.18 d. P(AB) =.12 Comment: the question asked about the second article ONLY

8 Problem # 12 A survey of 1000 randomly selected married women showed that 660 of these women have one or more children, while the rest have no children. Out of these 1000 women, 360 are working women; of the women who work, 250 have one or more children. A woman is selected randomly from this survey. Working Not working Total Have children No children Total a. What is the probability that this woman is i. a working woman = 360/1000 =.36 ii. a working woman given that she has one or more children = 250/660 =.38 iii. either a working woman or has no children = = 590 = b. Are the events ʺworking womanʺ and ʺhas no childrenʺ independent? Why or why not? P(Working) =.36 doesn t equal P(Working has no children) = 110/340 =.32. So the two events are dependent.

9 Problem # 13 A college student is planning his weekend (i.e., Thursday night and Friday night) activities. On Thursday night, he can sleep, go to a movie theater, or read. Each of these Thursday activities has an equal probability of occurrence. On Friday night, he can visit his family or do his homework. Each of these Friday activities has an equal probability of occurrence. Let S: sleep M: movie R: read Each with P = 1/3 V: visit H: homework Each with P = 1/2 a. Construct a tree diagram that would present the sequence of all possible weekend activities. S M R V H V H V H Outcome Probability SV 1/2.1/3= 1/6 SH 1/6 MV 1/6 MH 1/6 RV 1/6 RH 1/6 b. What is the probability that he will ultimately be doing his homework? P( he will ultimately be doing his homework) = P(SH) + P(MH) + P(RH) = 1/6 + 1/6 + 1/6 = 1/2

Chapter. Probability

Chapter. Probability Chapter 3 Probability Section 3.1 Basic Concepts of Probability Section 3.1 Objectives Identify the sample space of a probability experiment Identify simple events Use the Fundamental Counting Principle