Multiple Choice Exercise

*** Choose one answer and click [Submit] button

5.1 When two events A and B are mutually exclusive, what is the probability of an event A∪B?

P(A)－P(B)
P(A)P(B)
P(A)＋P(B)
P(A)/P(B)

        

5.2 Let the probability that event A will occur be P(A) and the probability that event B will occur be P(B). Which of the following is wrong?

0 ≤ P(A) ≤ 1
-1 ≤ P(B) ≤ 0
P(A∪B)=P(A)+P(B)-P(A∩B)
P(A∩B)=P(A)․P(B)

        

5.3 The value of P(A)=0.4, P(B)=0.2, P(A|B)=0.6. What is the probability of P(A∩B)?

0.08
0.12
0.24
0.48

        

5.4 If A ⊂ B, what is the comparison between the conditional probability P(A|B) and P(A)?

equal or greater
smaller
equal or smaller
There is no comparison

        

5.5 How likely are 2 and 5 to appear at the same time when throwing 2 dice?

1/3
1/6
1/12
1/18

        

5.6 When throwing a dice three times, what is the probability that the number of eyes is 5 at the first throw, 3 at the second, and an even number of eyes at the third?

1/30
1/72
1/108
1/276

        

5.7 When you extract three light bulbs randomly without replacement one by one from a barrel containing five good bulbs and two defective bulbs, what is the probability that one bulb is a defective item?

1/7
2/7
3/7.
4/7

        

5.8 When P(B)=0.2, P(A∩B)= 0.12, what is the value of P(A|B)?

0.20
0.60
0.12
0.24

        

5.9 When P(A)=0.4, P(B)=0.2, P(A|B)=0.6, what is the value of P(B|A)?

0.08
0.24
0.30
0.40

        

5.10 Mark the result of throwing two dice as ( \(x_1 , x_2 \) ) and let \(B = \{ (x_{1} ,x_{2} ) | x_{1} > x_{2} \} \) . What is the value of P(B)?

1/3
1/12
5/12
1/36

        

5.11 Mark the result of throwing two dice as ( \(x_1 , x_2 \) ) and let \(A = \{ (x_{1} ,x_{2} ) | x_{1} + x_{2} =10 \} \), \(B = \{ (x _{1} ,x _{2} ) | x _{1} > x _{2} \} \). What is the value of P(B|A)?

1/3
1/12
5/12
1/2

        

5.12 What is the standard deviation when we multiply each of the values of one random variable five times?

It is five times as much.
It is one-fifth the size.
It is ten times as much.
Four-tenths the size.

        

5.13 In a class, the average of mid-term scores was 24 points and its standard deviation was 3. Because of poor grades, professor doubled the mid-term scores and then add 10 points for all the students in the class. What is the average and standard deviation of new scores?

24, 3
48, 3
48, 6
58, 6

        

5.14 If the mean of a random variable \(\small X\) is 20, what is the mean of \(Y=2X+3\)?

20
40
23
43

        

5.15 If the variance of a random variable \(\small X\) is 2, what is the variance of \(Y=2X+1\)?

2
3
8
18

        

5.16 Which of the following is wrong?

\(\small E(aX＋b) = aE(X)＋b\)
\(\small E(X＋Y) = E(X)＋E(Y)\)
\(V(aX＋b) = a^2 V(X)\)
\(V(aX＋b) = a^2 V(X)+b\)

        

5.17 If the expected value of \(\small X\) is \(\small E(X)=5\) and \(\small E(X^2)\)=25, and what is the variance \(\small V(X)\)?

-5
0
5
25

        

5.18 Which one of the followings is suitable as a probability distribution function?

\(f(x) = \frac {x} {5} , \qquad(x = -1, 0, 1, 2, 3) \)
\(f(x) = \frac {(5-x)} {14} , \qquad(x = 0, 1, 2, 3) \)
\(f(x) = x^2 - 1 ,\qquad(x = 0, 1, 2, 3) \)
\(f(x) = x^3 - x^2 + 5 ,\qquad(x = 0, 1, 2, 3) \)

        

5.19 Which of the followings is NOT a probability distribution function?

\( f(x) \ge 0\)
\( \int_{-∞}^{∞} f(x) dx = 1 \)
\( P(a < X < b) = \int_a^b f(x)dx \)
\( \int_{-∞}^{∞} xf(x) dx = 1 \)

        

5.20 What is the average value of \(\small X\) when the variables \(\small X\) take values of 0, 1, 2 and 3 and the probability function \( f(x) = \frac{x}{6}\)

5/3
6/3
7/3
8/3

        

5.21 Which of the following is incorrect for the probability distribution function f(x) defined for all values of a continuous random variable \(\small X\)?

\( f(x) \ge 0\)
\( \int_{-∞}^{∞} f(x) dx = 1 \)
\( P(a < X < b) = \int_a^b f(x)dx \)
\( \int_{0}^{∞} f(x) dx = \frac{1}{2} \)

        

5.22 What is the mean value of \(\small E(X)\) and the standard deviation \(σ(X)\) when the probability distribution of a random variable \(\small X\) is as follows?

\(x\)	0	1	2	Total
\(\small P(X=x)\)	\(\frac{3}{10}\)	\(\frac{6}{10}\)	\(\frac{1}{10}\)	1

\(\small E(X)=0.8, σ(X)=0.6\)
\(\small E(X)=0.6, σ(X)=0.8\)
\(\small E(X)=0.7, σ(X)=0.1\)
\(\small E(X)=0.1, σ(X)=0.7\)

        

5.23 The number of eyes when we throw one dice is from 1 to 6. If \(\small X\) is a random variable which represents the number of eyes, what is the expected value of \(\small E(X)\)?

2
2.5
3
3.5

        

5.24 What is the expected value of scores if you square the number of eyes appeared when you throw one dice?

23/6
41/6
53/6
91/6

        

5.25 Assuming that a random variable \(\small X\) take three values 0, 1, and 2, and their probabilities are 1/2, 1/3, 1/6 respectively. What is the value of the cumulative distribution function F(x) if 1 ≤ x < 2?

0
5/6
1/3
2/3

        

5.26 What is the mean and variance of the binomial distribution \( B(100, \frac{1}{5} ) \)?

20, 4
100, 16
100, 4
20, 16

        

5.27 What is the variance of a random variable \(\small X\) which follows a binomial distribution \( P(X) = {}_4 C _x (0.2)^x (1-0.2) ^{4-x} \)

0.16
0.80
0.32
0.64

        

5.28 There are 10 multiple choice problems and only one of the four answers is the right answer. What is the probability that four questions are correct when answering 10 questions?

0.0162
0.0487
0.1460
0.2050

        

5.29 According to the nature of the normal distribution, what is the probability of taking a value between μ ± 3σ

Approximately 99.73%
Approximately 97.73%
Approximately 95.73%
Approximately 68.73%

        

5.30 There was a job test and the distribution of its scores is normal with the mean of 400 and standard deviation of 50. Scores between 450 and 500 are said to be suitable for the job. What percentage of scores do you think is appropriate for the job?

80%
about 50%
about 30%
about 14%

        

5.31 Using the standard normal distribution table \(P( -∞ < Z < z ) \) below, what is the probability of \(P(-0.41 \le Z \le 2.21)\)?

\(\small z\)	\(P( -∞ < Z < z ) \)
0.40	0.6554
0.41	0.6591
2.20	0.9861
2.21	0.9864

0.6415
0.6455
0.6452
0.6418

        

Practice Exercise

*** Answer the followings

5.1   Calculate the following.

5.2   When four offices are located side by side, they are run by four middle-level executives. In how many ways can these four executives be assigned to four offices?

5.3   A manager employs seven employees and four of them are to form a production team. In how many ways can seven people form different production teams?

5.4   An advertising designer wants to design pages of a magazine by selecting three of eight photos. Page location is said to be unimportant. How many designs can be made using different combinations of photos?

5.5   A manager of a company that produces five kinds of soap wants to display each sample soap in a row at a show case in his office. In how many ways can the manager display five soaps?

5.6   A salesperson has seven items and wants them on display at the counter. He can only display four. If the order in which he displays the item is not important, in how many ways can he display?

5.7   An airline company which has six airplanes plans to advertise each airplane's operation in the Sunday newspaper for six weeks.

5.8   A company said that five applicants have applied to fill five different positions. If all applicants are given equal qualifications for five positions, how many ways are there to fill five positions?

5.9   A company has 231 employees and employees are classified by age and rank as the following table.

	Age
Rank	A1 < 20	A2 21 - 25	A3 26 - 30	A4 31 - 35	A5 > 35	Total
B1 Clerk	20	20	15	10	5	70
B2 Manager	3	6	3	2	1	15
B3 Technician	15	30	35	20	10	110
B4 Salesman	1	5	10	5	2	23
B5 Director	0	1	5	2	0	8
B6 Executive	0	0	2	2	1	5
Total	39	62	70	41	19	231

5.10 A store which has three types of goods, A, B and C, has received orders from 200 customers for a certain period of time.

5.11 Two executive positions in a company are vacant. Seven men and three women are fully qualified and equally eligible. The company decided to pick two of these ten people at random. Find the following odds.

5.12 There are 300 households living in a neighborhood. One hundred of them are said to be out of the house in the evening. Of the remaining households, 50 do not respond to telephone surveys. When a person randomly selects a household in the neighborhood and conducts a telephone survey, ask for the following odds.

5.13 When an employee is selected randomly by a company, there is a 0.58 chance that the person is over the age of 31. What is the probability that an employee who is hired randomly is under 30?

5.14 The following table shows the number of absent days due to diseases of 100 workers over the year in a company. Answer the following questions based on this table.

\(\small X\) (Number of absent days)	0	1	2	3	4	5	6	7	8	9	10
Number of workers	5	8	10	12	18	14	10	9	8	4	2

5.15 When selecting one worker at random in question 5.14, obtain the following probabilities.

5.16 In question 5.14, obtain the following.

5.17 A department store surveyed 80 customers about their purchasing habits and asked, 'How many times did you buy at this department store last month?' The answer to the question is as follows.

\(\small X\)(Number of purchases)	0	1	2	3	4	5	6	7
Number of customers	15	27	14	12	6	4	1	1

5.18 Over the years, a salesperson found that there was a 50% chance of selling the product when a customer visited. If one day five customers visit this salesman, calculate the following odds. Check the calculation using 『eStatU』.

5.19 If a random variable \(\small X\) follows a binomial distribution with \(n = 6, p = 0.2\), find the following probabilities using 『eStatU』.

5.20 In some cities, 35% of residents are said to be opposed to the idea of expanding the main street. Obtain the following probabilities and check them using 『eStatU』.

5.21 It is said that 72% of drivers usually use a seat belt. When you select 15 drivers randomly and calculate the following probabilities of drivers who use a seat belt and check the calculation using 『eStatU』.

5.22 One producer claims that 6% of his product is defective. If you sample 20 products randomly, what is the probability that the number of defective products is as follows. Check the calculation using 『eStatU』.

5.23 The defect rate of products in a factory is said to be about 3%. When an employee continues to inspect the product until he finds a defective product to investigate the cause of defective products, find the following probabilities and check them using 『eStatU』.

5.24 The number of defects per 1 \(m^2\)in a fabric follows a Poisson distribution with the average defect = 0.2. When 1 \(m^2\)of this fabric is investigated for quality inspection, find the following probabilities and check the calculation using 『eStatU』.

5.25 Assume that the average number of Hurricanes passing through the southern part of the country is = 3 times per year. Check the following probabilities using 『eStatU』.

5.26 A candidate has a 60% approval rating in an election. When interviewing voters to hear the opinions of those who oppose the candidate, look for the next probabilities.

5.27 Five samples are selected randomly from a box containing 20 tobacco products of which there are 15 normal products and 5 defective products. What is the probability of having one, two, or three defectives in the samples?

5.28 If \(\small Z\) is the standard normal random variable, obtain the following and check the calculation using 『eStatU』.

5.29 It is said that a tin can lid produced by a company follows a normal distribution with the average diameter of 4 and the standard deviation of 0.012. Calculate the probability that the lids are between 3.97 and 4.03. Check the calculation using 『eStatU』.

5.30 It is said that the weight of a melon follows a normal distribution with the average of 250g and standard deviation of 12g. Find the probability that the weight of the melon is less than 260 grams. Check the calculation using 『eStatU』.

5.31 A bank employee found that the length of waiting time of a customer to receive a service follows a normal distribution with the average of 3 minutes and the standard deviation of 1 minute. Calculate the following probabilities and check the calculation using 『eStatU』.

5.32 It takes an average of 10 minutes for a production worker to finish a job at a factory and time follows a normal distribution with a standard deviation of 3 minutes. Calculate the following probabilities and check the calculation using 『eStatU』.

5.33 An average product has a life expectancy of 100 hours and follows an exponential distribution. Calculate the following probabilities and check the calculation using 『eStatU』.