Probability and Chance: Interactive Learning Quizzes for Advanced Diploma in Engineering

Question 1

A bag contains 4 red balls, 3 blue balls, and 5 yellow balls. If one ball is drawn randomly from the bag, what is the probability of drawing a blue ball?

Accepted Answer

1/4

Answer

5/12

Answer

1/3

Answer

3/12

Question 2

Which of the following is NOT a property of probability?

Accepted Answer

Probability can be negative.

Answer

Probability is always between 0 and 1 (inclusive).

Answer

The sum of probabilities of all possible outcomes in a sample space is always 1.

Answer

Probability is a measure of uncertainty.

Question 3

Bayes' theorem is primarily used to:

Accepted Answer

Calculate the probability of an event given that another event has already occurred.

Answer

Calculate the probability of two independent events occurring.

Answer

Calculate the probability of an event based on its complement.

Question 4

A bag contains 5 red balls and 3 blue balls. If a ball is drawn at random, what is the probability that it is red?

Accepted Answer

5/8

Answer

5/3

Answer

3/8

Answer

3/5

Question 5

Two events A and B are mutually exclusive. If P(A) = 0.3 and P(B) = 0.4, what is P(A or B)?

Accepted Answer

0.7

Answer

0.58

Answer

1

Answer

0.12

Question 6

The probability of event A happening given that event B has already occurred is 0.6. If P(B) = 0.4, what is P(A and B)?

Accepted Answer

0.24

Answer

0.14

Answer

1

Answer

0.67

Question 7

Which of the following is NOT a property of a probability density function f(x) for a continuous random variable X?

Accepted Answer

f(x) can be greater than 1 for some values of x

Answer

P(a < X < b) is represented by the area under the curve of f(x) between a and b

Answer

The total area under the curve of f(x) is equal to 1

Answer

f(x) ≥ 0 for all x

Question 8

A random variable X has a normal distribution with a mean of 50 and a standard deviation of 5. What is the approximate value of P(X > 55)?

Accepted Answer

0.1587

Answer

0.6587

Answer

0.3413

Answer

0.8413

Question 9

The expected value of a discrete random variable X is:

Accepted Answer

The weighted average of all possible values of X, where the weights are the probabilities of each value.

Answer

The most likely value of X.

Question 10

If two events A and B are independent, then:

Accepted Answer

P(A|B) = P(A)

Answer

P(A) = P(B)

Answer

P(A and B) = P(A) + P(B)

Question 11

A company wants to estimate the average number of hours its employees spend on training each year. They randomly select 50 employees and find their average training time is 20 hours, with a standard deviation of 5 hours.  What is the 95% confidence interval for the average training time of all employees?

Accepted Answer

(18.63, 21.37)

Answer

(19.37, 20.63)

Answer

(15, 25)

Answer

(10, 30)

Question 12

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If you randomly select one ball, what is the probability of drawing a blue ball?

Accepted Answer

3/10

Answer

1/5

Answer

1/2

Answer

2/5

Question 13

A fair coin is flipped three times. What is the probability of getting at least one head?

Accepted Answer

7/8

Answer

3/4

Answer

1/8

Answer

1/2

Question 14

A box contains 4 defective light bulbs and 6 good light bulbs. If you randomly select two bulbs without putting the first one back, what is the probability that both bulbs are defective?

Accepted Answer

1/15

Answer

2/15

Answer

1/5

Answer

1/10

Question 15

A company produces two types of batteries, A and B. Battery A has a 90% success rate, and Battery B has a 75% success rate. 60% of the batteries produced are type A. If a battery is randomly selected and found to be successful, what is the probability that it is type A?

Accepted Answer

0.72

Answer

0.90

Answer

0.60

Answer

0.48

Question 16

A random variable X represents the number of heads obtained when a fair coin is flipped four times. What is the expected value of X?

Accepted Answer

2

Answer

1

Answer

3

Question 17

A machine produces bolts with a mean length of 10 cm and a standard deviation of 0.5 cm. Assuming the bolt lengths are normally distributed, what is the probability that a randomly selected bolt will have a length between 9.5 cm and 10.5 cm?

Accepted Answer

0.6827

Answer

0.9545

Answer

0.3413

Answer

0.1587

Question 18

A company claims that 95% of its products meet quality standards. To verify this claim, a random sample of 100 products is inspected. What is the probability that less than 90% of the products in the sample will meet the quality standards, assuming the company's claim is true?

Accepted Answer

Less than 0.05

Answer

Cannot be determined from the information given

Answer

Greater than 0.95

Answer

Approximately 0.5

Question 19

The probability distribution of a discrete random variable X is given in the table below. What is the variance of X?

| X | 0 | 1 | 2 | 3 | 
|---|---|---|---|---| 
| P(X) | 0.2 | 0.3 | 0.4 | 0.1 |

Accepted Answer

1.1

Answer

0.8

Answer

2.0

Answer

1.5

Question 20

A system has three components connected in series. The probabilities of each component failing are 0.1, 0.2, and 0.3, respectively. What is the probability that the entire system will work successfully?

Accepted Answer

0.504

Answer

0.600

Answer

0.700

Answer

0.900

Question 21

A company uses a machine to produce widgets. The machine produces an average of 5 defective widgets per 100. What is the probability that a batch of 20 widgets will contain at least one defective widget?

Accepted Answer

0.633

Answer

0.05

Answer

0.367

Answer

0.95

Question 22

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If a ball is drawn at random, what is the probability of drawing a blue ball?

Accepted Answer

3/10

Answer

1/3

Answer

1/2

Answer

2/5

Question 23

A fair coin is tossed twice. What is the probability of getting at least one head?

Accepted Answer

3/4

Answer

2/3

Answer

1/2

Answer

1/4

Question 24

A box contains 4 defective and 6 non-defective light bulbs. If 2 bulbs are randomly selected without replacement, what is the probability that both are defective?

Accepted Answer

1/15

Answer

1/10

Answer

2/15

Answer

1/5

Question 25

A factory produces 80% of its products in one plant and 20% in another. Plant A has a 1% defective rate, while Plant B has a 3% defective rate. If a randomly selected product is found to be defective, what is the probability it was produced in Plant B?

Accepted Answer

0.375

Answer

0.15

Answer

0.2

Answer

0.625

Question 26

A random variable X represents the number of heads obtained when flipping a fair coin three times. What is the expected value of X?

Accepted Answer

1.5

Answer

2

Answer

1

Question 27

A machine produces bolts with lengths that are normally distributed with a mean of 10 cm and a standard deviation of 0.5 cm. What is the probability that a randomly selected bolt will have a length between 9.5 cm and 10.5 cm?

Accepted Answer

0.6827

Answer

0.9545

Answer

0.1587

Answer

0.3413

Question 28

A company manufactures components with a 2% defect rate. What is the probability of finding at least one defective component in a sample of 100 components?

Accepted Answer

0.867

Answer

0.98

Answer

0.02

Answer

0.133

Question 29

A survey found that 60% of engineers prefer programming language A, while 40% prefer language B. If 10 engineers are randomly selected, what is the probability that exactly 6 of them prefer language A?

Accepted Answer

0.2508

Answer

0.4

Answer

0.6

Answer

0.1

Question 30

The probability distribution of a discrete random variable X is given below: 
    | X | 0 | 1 | 2 | 3 |
    |---|---|---|---|---|
    | P(X) | 0.2 | 0.3 | 0.4 | 0.1 |
    What is the variance of X?

Accepted Answer

1.06

Answer

0.84

Answer

1.5

Answer

2.12

Question 31

A company is developing a new type of sensor. The probability of success for each trial is 0.7. What is the probability that the company will require at least 3 trials to achieve success?

Accepted Answer

0.027

Answer

0.973

Answer

0.7

Answer

0.27

Question 32

A bag contains 3 red balls, 4 blue balls, and 5 green balls. If two balls are drawn at random from the bag, without replacement, what is the probability of drawing a red ball followed by a green ball?

Accepted Answer

9/55

Answer

3/22

Answer

1/10

Answer

12/55

Question 33

A fair coin is tossed twice. What is the probability of getting heads on the first toss and tails on the second toss?

Accepted Answer

1/4

Answer

1/8

Answer

1/16

Answer

1/2

Question 34

A survey found that 60% of students like Math and 40% like Science. If 30% of students like both Math and Science, what is the probability that a randomly selected student likes only Math?

Accepted Answer

42%

Answer

52%

Answer

62%

Answer

32%

Question 35

A random variable X has the following probability distribution: 
 X: 1 2 3 
P(X): 0.4 0.3 0.3 
What is the expected value of X?

Accepted Answer

2

Answer

4

Answer

3

Answer

1

Question 36

A manufacturing process produces items that are either defective or non-defective. The probability of producing a defective item is 0.1. If a random sample of 10 items is selected, what is the probability of getting exactly 2 defective items?

Accepted Answer

0.246

Answer

0.1

Answer

0.36

Answer

0.496

Question 37

Bayes' theorem is used to calculate:

Accepted Answer

Conditional probability based on new evidence

Answer

Variance

Answer

Expected value

Answer

Independent probability

Question 38

In a population of 1000 people, 450 are male and 550 are female. 250 males and 300 females have a certain disease. What is the probability that a randomly selected person from the population has the disease?

Accepted Answer

0.275

Answer

0.375

Answer

0.475

Question 39

A bag contains 6 red balls and 4 blue balls. Three balls are drawn at random, with replacement. What is the probability of drawing 2 red balls and 1 blue ball?

Accepted Answer

108/216

Answer

162/216

Answer

54/216

Answer

64/216

Question 40

A continuous random variable X has the following probability density function: 
f(x) = 2x 
for 0 ≤ x ≤ 1 
What is the probability that X takes a value between 0.2 and 0.8?

Accepted Answer

0.6

Answer

0.8

Answer

0.4

Question 41

A fair six-sided die is rolled twice. What is the probability of getting a sum of 7?

Accepted Answer

1/6

Answer

1/18

Answer

1/12