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Test-1 ( First Mid )- to be active on 08/09/2024 at 3.00 PM

Multiple Choice Questions & Notes

Practice the MCQs & Attempt the Tests

1. What is the probability of getting a head when flipping a fair coin?

• a) 0
• b) 0.25
• c) 0.5
• d) 1
• Answer: c) 0.5

2. A die is rolled. What is the probability of getting a number less than 4?

• a) 1/3
• b) 1/2
• c) 2/3
• d) 3/6
• Answer: c) 2/3

3. If two events are mutually exclusive, what is the probability of their union?

• a) P(A) * P(B)
• b) P(A) + P(B)
• c) P(A) + P(B) – P(A∩B)
• d) P(A) – P(B)
• Answer: b) P(A) + P(B)

4. Which of the following is an example of a continuous sample space?

• a) Rolling a die
• b) Flipping a coin
• c) Measuring the height of students
• d) Counting the number of heads in 3 coin tosses
• Answer: c) Measuring the height of students

5. In a set of 100 trials, an event occurs 30 times. What is the relative frequency of the event?

• a) 0.3
• b) 0.7
• c) 0.03
• d) 0.333
• Answer: a) 0.3

6. What is the probability of drawing an ace from a standard deck of cards?

• a) 1/13
• b) 1/52
• c) 4/52
• d) 1/4
• Answer: a) 1/13

7. If two events A and B are independent, what is the probability of both occurring?

• a) P(A) + P(B)
• b) P(A) – P(B)
• c) P(A) * P(B)
• d) P(A) / P(B)
• Answer: c) P(A) * P(B)

8. Bayes’ Theorem is used to calculate:

• a) The probability of an event
• b) The conditional probability
• c) The relative frequency
• d) The joint probability
• Answer: b) The conditional probability

9. The sum of all probabilities in a sample space is:

• a) 0
• b) 0.5
• c) 1
• d) Depends on the sample space
• Answer: c) 1

10. What is the probability of rolling an even number on a fair 6-sided die?

• a) 1/6
• b) 1/3
• c) 1/2
• d) 2/3
• Answer: c) 1/2

11. Which of the following is not a property of a probability distribution?

• a) The probability of any event is between 0 and 1.
• b) The sum of the probabilities of all possible events is 1.
• c) Probability is negative for some events.
• d) The probability of an impossible event is 0.
• Answer: c) Probability is negative for some events.

12. What is the complement of event A?

• a) A∩B
• b) A∪B
• c) Not A
• d) A – B
• Answer: c) Not A

13. In a binomial experiment, the probability of exactly one success in three trials is given by:

• a) 3p(1-p)²
• b) 3p²(1-p)
• c) p(1-p)
• d) p²(1-p)²
• Answer: a) 3p(1-p)²

14. If two events A and B are independent, what is the probability of A given B?

• a) P(A)
• b) P(B)
• c) P(A) * P(B)
• d) P(A|B)
• Answer: a) P(A)

15. The probability of getting exactly 2 heads in 3 tosses of a fair coin is:

• a) 1/2
• b) 1/3
• c) 1/4
• d) 3/8
• Answer: d) 3/8

16. If A and B are mutually exclusive events, what is the value of P(A∩B)?

• a) 1
• b) P(A)
• c) P(B)
• d) 0
• Answer: d) 0

17. If the probability of event A is 0.4 and the probability of event B is 0.5, what is the probability of both events A and B occurring, assuming they are independent?

• a) 0.2
• b) 0.9
• c) 0.45
• d) 0.4
• Answer: a) 0.2

18. What is the sample space when a die is rolled?

• a) {1, 2, 3, 4, 5, 6}
• b) {1, 2, 3, 4}
• c) {H, T}
• d) {Even, Odd}
• Answer: a) {1, 2, 3, 4, 5, 6}

19. The probability that a randomly chosen person has a birthday on February 29 is:

• a) 1/365
• b) 1/366
• c) 1/1461
• d) 1/730
• Answer: c) 1/1461

20. If an event A has a probability of 0.7, what is the probability of its complement?

• a) 0.3
• b) 0.7
• c) 1
• d) 0.5
• Answer: a) 0.3

21. The set of all possible outcomes of an experiment is called the:

• a) Event space
• b) Sample space
• c) Outcome space
• d) Probability space
• Answer: b) Sample space

22. The mathematical model used in probability theory is called:

• a) Sample space
• b) Event space
• c) Probability space
• d) Statistical space
• Answer: c) Probability space

23. What is the probability of rolling a 7 on a standard 6-sided die?

• a) 1/6
• b) 1/7
• c) 0
• d) 1/12
• Answer: c) 0

24. If event A is certain to occur, what is P(A)?

• a) 0
• b) 0.5
• c) 1
• d) 0.75
• Answer: c) 1

25. The probability of the intersection of two independent events is given by:

• a) P(A) + P(B)
• b) P(A) * P(B)
• c) P(A) – P(B)
• d) P(A) / P(B)
• Answer: b) P(A) * P(B)

26. What does it mean if two events are independent?

• a) The occurrence of one event affects the occurrence of the other.
• b) The occurrence of one event does not affect the occurrence of the other.
• c) The events cannot happen at the same time.
• d) The events are mutually exclusive.
• Answer: b) The occurrence of one event does not affect the occurrence of the other.

27. What is the probability that a randomly selected day of the week is a Monday?

• a) 1/7
• b) 2/7
• c) 3/7
• d) 1/5
• Answer: a) 1/7

28. In a deck of 52 cards, what is the probability of drawing a heart?

• a) 1/4
• b) 1/13
• c) 1/52
• d) 13/52
• Answer: a) 1/4

• Topics Covered:
• Probability theory (including sets, relative frequency, etc.)
• Stochastic processes (including Markov processes, Poisson processes, etc.)
• Questions (with varying difficulty levels)
• What is the probability of getting a head when flipping a fair coin?
  • a) 0.25
  • b) 0.5
  • c) 0.75
  • d) 1
  • Answer: b) 0.5
• In a probability experiment, what is the set of all possible outcomes called?
  • a) Event
  • b) Sample space
  • c) Experiment
  • d) Trial
  • Answer: b) Sample space
• If two events A and B are mutually exclusive, what is P(A ∪ B)?
  • a) P(A) + P(B)
  • b) P(A) * P(B)
  • c) P(A) – P(B)
  • d) P(A) / P(B)
  • Answer: a) P(A) + P(B)
• A random experiment has a discrete sample space. Which of the following is a valid probability distribution?
  • a) Sum of all probabilities is greater than 1
  • b) Sum of all probabilities is less than 1
  • c) Sum of all probabilities is exactly 1
  • d) Probabilities are negative
  • Answer: c) Sum of all probabilities is exactly 1
• Bayes’ Theorem helps to compute which of the following?
  • a) Joint probability
  • b) Marginal probability
  • c) Conditional probability
  • d) Independent probability
  • Answer: c) Conditional probability
• In a Poisson process, the number of events occurring in a fixed interval of time is:
  • a) Constant
  • b) Normally distributed
  • c) Poisson distributed
  • d) Exponentially distributed
  • Answer: c) Poisson distributed
• Which of the following processes is characterized by the “memoryless” property?
  • a) Markov process
  • b) Renewal process
  • c) Brownian motion
  • d) Stationary process
  • Answer: a) Markov process
• What is the probability of drawing an ace from a standard deck of 52 cards?
  • a) 1/52
  • b) 1/26
  • c) 1/13
  • d) 1/4
  • Answer: c) 1/13
• For two independent events A and B, the probability of both occurring is:
  • a) P(A) + P(B)
  • b) P(A) * P(B)
  • c) P(A) – P(B)
  • d) P(A) / P(B)
  • Answer: b) P(A) * P(B)
• Which of the following represents the autocorrelation function of a wide-sense stationary process?
  • a) A constant value
  • b) A function of time difference
  • c) A linear function of time
  • d) A random variable
  • Answer: b) A function of time difference
• Which of the following is a characteristic of a Poisson process?
  • a) Events are clustered
  • b) Events are independent
  • c) Events are dependent
  • d) Events are deterministic
  • Answer: b) Events are independent
• In a discrete probability distribution, what is the probability that a random variable X takes a specific value x?
  • a) P(X < x)
  • b) P(X = x)
  • c) P(X > x)
  • d) P(X ≠ x)
  • Answer: b) P(X = x)
• The expected value of a random variable is also known as:
  • a) Mode
  • b) Median
  • c) Mean
  • d) Variance
  • Answer: c) Mean
• In a continuous probability distribution, the probability that a random variable equals a specific value is:
  • a) Zero
  • b) One
  • c) Infinity
  • d) A positive finite number
  • Answer: a) Zero
• Which of the following is true for independent events?
  • a) P(A | B) = P(A)
  • b) P(A ∩ B) = 0
  • c) P(A ∩ B) = P(A) + P(B)
  • d) P(A | B) = P(A ∩ B)
  • Answer: a) P(A | B) = P(A)
• A random process where future states are independent of past states given the present state is called:
  • a) Stationary process
  • b) Markov process
  • c) Poisson process
  • d) Renewal process
  • Answer: b) Markov process
• Which theorem provides a way to calculate the total probability of an event based on conditional probabilities?
  • a) Central Limit Theorem
  • b) Law of Large Numbers
  • c) Bayes’ Theorem
  • d) Total Probability Theorem
  • Answer: d) Total Probability Theorem
• Which stochastic process is used to model queueing systems?
  • a) Brownian motion
  • b) Markov chains
  • c) Poisson process
  • d) Renewal process
  • Answer: c) Poisson process
• In which process does the next state depend only on the current state and not on the sequence of events that preceded it?
  • a) Markov process
  • b) Stationary process
  • c) Renewal process
  • d) Random walk
  • Answer: a) Markov process
• Which of the following measures the strength and direction of a linear relationship between two random variables?