Assignment 2 (Solution) Probability and Statistics

Transcription

1 Assignment 2 (Solution) Probability and Statistics Dr. Jitesh J. Thakkar Department of Industrial and Systems Engineering Indian Institute of Technology Kharagpur

2 Instruction Total No. of Questions: 15. Each question carries one point. All questions are objective type. In some of the questions, more than one answers are correct. This assignment includes true/false statement questions.

3 Question 1 Bag I contains 6 white and 4 black balls while another Bag II contains 3 white and 4 black balls. One ball is drawn at random from one of the bags and it is found to be black. What is the probability that it was drawn from Bag II? a) b) c) d) 0.644

4 Answer: c (0.588) Solution 1 Let E 1 be the event of choosing the bag I, E 2 the event of choosing the bag II and A be the event of drawing a black ball. Then, P(E 1 ) = P(E 2 ) = 1 2 Also, P(A E 1 ) = P(drawing a black ball from Bag I) = 4 10 = 2 5 P(A E 2 ) = P(drawing a black ball from Bag II) = 4 7 By using Bayes theorem, the probability of drawing a black ball from bag II out of two bags, P(E 2 A) = 1 P(E 2 )P(A E 2 ) P(E 1 )P(A E 1 )+P(E 2 )P(A E 2 ) = = = 0.588

5 Match the following. Question 2 1. Events a. Head will be mapped to 1, and tail will be mapped to 0, in Coin tossing. 2. Sample Space b. The possible outcomes of a stochastic or random process 3. Random Variable c. The set that consists of all the outcomes a) 1-c, 2-b, 3-a b) 1-b, 2-a, 3-c c) 1-b, 2-c, 3-a d) 1-c, 2-a, 3-b

6 Solution 2 Answer: c (1-b, 2-c, 3-a) 1. Events b. The possible outcomes of a stochastic or random process 2. Sample Space c. The set that consists of all the outcomes 3. Random Variable a. Head will be mapped to 1, and tail will be mapped to 0, in Coin tossing.

7 a) 0.22 b) 0.34 c) 0.45 d) None of these Question 3 At Cornell School, all first year students must take chemistry and math. Suppose 25% fail in chemistry, 18% fail in math, and 9% fail in both. Suppose a first year student is selected at random. What is the probability that student selected failed at least one of the courses?

8 Answer: b (0.34) Solution 3 To Solve this problem use Additive law of Probability. P A B = P A + P B P(A B) P(C) = 0.25 P(M) = 0.18 P C M = 0.09 P(at least one) = P(C or M) = P(C M) = = 0.34

9 Question 4 Which of the following statement is correct? a) The probability of rain today is 23%. b) The probability of rain today is -10%. c) The probability of rain today is 120%. d) The probability of rain or no rain today is 80%.

10 Solution 4 Answer: a (The probability of rain today is 23%) Remember!!!! P an event 0 P an event 1 P Sample Set = 1

11 The given table shows the data of Two-way Tables and Probability of a class. Find the probability that the student getting Ex grade is a girl. a) b) 4 9 Question 5 Got Ex Got < Ex Girl Boy c) 5 36 d) 5 13

12 Answer: d ( 5 13 ) Total Boy = 115 Total Girl = 65 Solution 5 Probability that the student is a girl =P(G)= Total Ex grade holders = 80 Probability that a student is a Ex grade holder = P(A) = 80/180 The number of girls getting Ex grade = 25 Probability that a student getting Ex grade and a girl = P(A G) = Probability that a student getting Ex grade is a girl = P(A G) = P(A G) = 25/180 = 25 = 5 P(G) 65/

13 Question 6 A pack contains 4 blue, 2 red and 3 black pens. If 2 pens are drawn at random from the pack, NOT replaced and then another pen is drawn. What is the probability of drawing 2 blue pens and 1 black pen? a) 1 14 b) 2 14 c) 3 14 d) 4 14

14 Answer: a ( 1 14 ) Solution 6 Probability of drawing 1 blue pen = 4/9 Probability of drawing another blue pen = 3/8 Probability of drawing 1 black pen = 3/7 Probability of drawing 2 blue pens and 1 black pen = 4/9 * 3/8 * 3/7 = 1/14

15 Question 7 If two events are Mutually Exclusive then which of the following holds good? a) Events whose occurrence do not depend on the occurrence of any other events b) The occurrence of one precludes the occurrence of the other c) P A B = 0 d) P A B = P A + P(B)

16 Solution 7 Answer: b and c [P A B = 0] Examples of Mutually Exclusive Events: 1. In a soccer game scoring no goal (event A) and scoring exactly one goal (event B). 2. In a deck of cards Kings (event A) and Queens (event B). Examples Not Mutually Exclusive Events: In a deck of cards Kings (event A) and Hearts (event B). They have common cards i.e. Heart King. So A B 0. Events whose occurrence do not depend on the occurrence of any other events are called independent events.

17 Question 8 If a coin is tossed five times then what is the probability that you observe at least one head? a) 1 32 b) c) d) 31 32

18 Answer: d ( ) Solution 8 Consider solving this using complement. Probability of getting no head = P(all tails) = 1/32 P(at least one head) = 1 P(all tails) = 1 1/32 = 31/32.

19 Question 9 A desk lamp produced by The Luminar Company was found to be defective (D). There are three factories (A, B, C) where such desk lamps are manufactured. A Quality Control Manager (QCM) is responsible for investigating the source of found defects. This is what the QCM knows about the company's desk lamp production and the possible source of defects: Factory % of total production Probability of defective lamps A 0.35 = P(A) = P(D A) B 0.35 = P(B) = P(D B) C 0.30 = P(C) = P(D C)

20 Question 9 The QCM would like to answer the following question: If a randomly selected lamp is defective, what is the probability that the lamp was manufactured in factory B? a) b) c) d) 0.237

21 Solution 9 Answer: d (0.237) Using conditional probability and Bay s Theorem P(B D)= P(B D) P(D) = = = P D B.P(B) P(D) P D B. P(B) P[(D A) (D B) (D C)] P D B. P(B) P D A. P A + P D B. P B + P D C. P(C) (0.01). (0.35) = (0.015). (0.35) + (0.01). (0.35) + (0.02). (0.30) = 0.237

22 Question 10 One card is randomly drawn from a pack of 52 cards. What is the probability that the card drawn is a face card(jack, Queen or King)? a) 1 13 b) 2 13 c) 3 13 d) 4 13

23 Solution 10 Answer: c ( 3 13 ) Total number of cards, n(s) = 52 Total number of face cards, n(e) = 12 P(E) = n(e) n(s) = = 3 13

24 Question 11 Consider the experiment of tossing a coin twice. A is an event having sample set as {HT,HH} and B is an event having sample set {HT}. Event A is independent from Event B. a) True b) False

25 Solution 11 Answer: b (False) When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring.

26 Question 12 In a class, 40% of the students study math and science. 60% of the students study math. What is the probability of a student studying science given he/she is already studying math? a) 0.34 b) 0.53 c) 0.67 d) 0.74

27 Solution 12 Answer: c (0.67) P(M and S) = 0.40 P(M) = 0.60 P(S M) = P(M and S)/P(M) = 0.40/0.60 = 2/3 = 0.67

28 Question 13 At a car park there are 100 vehicles, 60 of which are cars, 30 are vans and the remainder are Lorries. If every vehicle is equally likely to leave, find the probability of car leaving second if either a lorry or van had left first. a) 3 10 b) c) 1 10 d) 3 5

29 Solution 13 Answer: b ( ) Let S be the sample space and A be the event of a van leaving first. n(s) = 100, n(a) = 30 Probability of a van leaving first: P(A) = 30 = Let B be the event of a lorry leaving first. n(b) = = 10 Probability of a lorry leaving first: P(B) = = 1 10 If either a lorry or van had left first, then there would be 99 vehicles remaining, 60 of which are cars. Let T be the sample space and C be the event of a car leaving second. n(t) = 99, n(c) = 60 Probability of a car leaving after a lorry or van has left: P(B) = 60 =

30 Question 14 What is the formula to find skewness of a distribution? n i 1 (x x i ) 3 a) g = Where x is the (n 1)s 3 observation, x i is the mean, n is the total number of observations and s is the variance. b) Skewness = 3 (mean median) standard deviation c) Both a and b d) None of these

31 Solution 14 Answer: c (Both a and b) n (x x i ) 3 i 1 Skewness(g) = Where x is the (n 1)s 3 observation, x i is the mean, n is the total number of observations and s is the variance. The formula to find skewness manually Skewness = 3 (mean median) standard deviation We have the following rules as per Bulmer in the year 1979: If the skewness comes to less than -1 or greater than +1, the data distribution is highly skewed If the skewness comes to between -1 and 1 or between skewed. and +1, the data distribution is moderately If the skewness is between 1 and + 1, the distribution is approximately symmetric 2 2

32 Question 15 What is the median and mode for the following list of values? 13, 18, 13, 14, 13, 16, 14, 21, 13 a) 14 and 13 b) 15 and 16 c) 14 and 15 d) 13 and 16

33 Solution 15 Answer: a (14 and 13) The median is the middle value, so to rewrite the list in order: 13, 13, 13, 13, 14, 14, 16, 18, 21 There are nine numbers in the list, so the middle one will be =5 = 5th number, So the median is 14. The mode is the number that is repeated more often than any other, so 13 is the mode.

34 Thank you