Probability theory, Mcqs

1. What is the probability of an event that is certain to happen?

a) 0
b) 0.5
c) 1
d) None of the above
Answer: c) 1

2. If two events are mutually exclusive, what is the probability that either one or the other occurs?

a) Sum of their probabilities
b) Product of their probabilities
c) Difference of their probabilities
d) Zero
Answer: a) Sum of their probabilities

3. What is the probability of an impossible event?

a) 1
b) 0.5
c) 0
d) None of the above
Answer: c) 0

4. Which of the following is not a valid probability value?

a) 0.75
b) -0.25
c) 0.5
d) 1
Answer: b) -0.25

5. If two events A and B are independent, what is the probability that both A and B occur?

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)

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

a) 1
b) 0.75
c) 0.5
d) 0.25
Answer: c) 0.5

7. What is the sum of probabilities of all possible outcomes of a random experiment?

a) 0
b) 0.5
c) 1
d) Depends on the experiment
Answer: c) 1

8. In a standard deck of cards, what is the probability of drawing an ace?

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

9. Which theorem is used to find the probability of the intersection of two events?

a) Bayes’ Theorem
b) Addition Theorem
c) Multiplication Theorem
d) Central Limit Theorem
Answer: c) Multiplication Theorem

10. If two events are mutually exclusive, what is the probability that they both occur?

a) 0
b) 0.5
c) 1
d) Cannot be determined
Answer: a) 0

11. Which of the following is an example of a continuous random variable?

a) Number of students in a class
b) Roll of a die
c) Height of students
d) Number of heads in 10 coin flips
Answer: c) Height of students

12. The probability that event A occurs given that event B has occurred is called:

a) Marginal probability
b) Joint probability
c) Conditional probability
d) Independent probability
Answer: c) Conditional probability

13. If P(A) = 0.3 and P(B) = 0.4, what is P(A and B) if A and B are independent?

a) 0.12
b) 0.1
c) 0.7
d) 0.0
Answer: a) 0.12

14. Which distribution describes the number of successes in a fixed number of trials?

a) Binomial distribution
b) Normal distribution
c) Poisson distribution
d) Exponential distribution
Answer: a) Binomial distribution

15. What is the expected value of a random variable X with probability mass function P(X=x) = x/15 for x=1,2,3?

a) 2
b) 1.5
c) 3
d) 4
Answer: a) 2

16. Which theorem states that the sum of a large number of random variables will be approximately normally distributed?

a) Bayes’ Theorem
b) Central Limit Theorem
c) Law of Large Numbers
d) Chebyshev’s Theorem
Answer: b) Central Limit Theorem

17. Which measure is used to describe the spread of a probability distribution?

a) Mean
b) Median
c) Variance
d) Mode
Answer: c) Variance

18. What is the variance of a fair six-sided die roll?

a) 2.5
b) 3.5
c) 5.5
d) 35
Answer: a) 2.5

19. Which distribution is appropriate for modeling the number of events occurring in a fixed interval of time or space?

a) Binomial distribution
b) Poisson distribution
c) Exponential distribution
d) Uniform distribution
Answer: b) Poisson distribution

20. If X is a normal random variable with mean μ and standard deviation σ, what is the probability that X is within one standard deviation of the mean?

a) 0.68
b) 0.95
c) 0.99
d) 0.5
Answer: a) 0.68

21. Which distribution has a bell-shaped probability density function?

a) Binomial distribution
b) Poisson distribution
c) Normal distribution
d) Exponential distribution
Answer: c) Normal distribution

22. What is the standard deviation of a binomial distribution with parameters n and p?

a) √(np(1-p))
b) np
c) n(1-p)
d) p(1-p)
Answer: a) √(np(1-p))

23. Which of the following is a measure of central tendency?

a) Variance
b) Standard deviation
c) Median
d) Range
Answer: c) Median

24. What is the probability that a continuous random variable takes on a specific value?

a) 1
b) 0.5
c) 0
d) Depends on the distribution
Answer: c) 0

25. Which theorem provides a way to update probabilities based on new evidence?

a) Central Limit Theorem
b) Bayes’ Theorem
c) Law of Large Numbers
d) Chebyshev’s Theorem
Answer: b) Bayes’ Theorem

26. What is the mean of a uniform distribution on the interval [a, b]?

a) (a + b)/2
b) (a – b)/2
c) (a + b)/4
d) (a – b)/4
Answer: a) (a + b)/2

27. Which distribution is used to model the time between successive events?

a) Binomial distribution
b) Poisson distribution
c) Exponential distribution
d) Normal distribution
Answer: c) Exponential distribution

28. If two events A and B are independent, what is the probability that A or B occurs?

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

29. What is the variance of a normal distribution with mean μ and standard deviation σ?

a) μ
b) σ
c) σ^2
d) μ^2
Answer: c) σ^2

30. Which distribution describes the number of trials until the first success?

a) Binomial distribution
b) Poisson distribution
c) Geometric distribution
d) Exponential distribution
Answer: c) Geometric distribution

31. In a normal distribution, what percentage of data falls within two standard deviations of the mean?

a) 68%
b) 95%
c) 99%
d) 50%
Answer: b) 95%

32. What is the expected value of a discrete random variable?

a) The probability of each value
b) The sum of all possible values
c) The weighted average of all possible values
d) The highest possible value
Answer: c) The weighted average of all possible values

33. If P(A|B) = P(A), what can be said about events A and B?

a) They are mutually exclusive
b) They are independent
c) They are dependent
d) They are conditional
Answer: b) They are independent

34. Which measure is used to describe the symmetry of a probability distribution?

a) Mean
b) Median
c) Skewness
d) Kurtosis
Answer: c) Skewness

35. What is the probability of getting exactly two heads in three flips of a fair coin?

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

36. Which distribution is used to model binary outcomes?

a) Normal distribution
b) Binomial distribution
c) Exponential distribution
d) Uniform distribution
Answer: b) Binomial distribution

37. What is the probability density function (PDF) of a continuous random variable?

a) A function that gives the probability that the variable takes a specific value
b) A function that gives the cumulative probability
c) A function that gives the likelihood of the variable being less than or equal to a value
d) A function that gives the likelihood of the variable being greater than or equal to a value
Answer: b) A function that gives the cumulative probability

38. Which of the following distributions has a parameter called the rate?

a) Poisson distribution
b) Geometric distribution
c) Exponential distribution
d) Uniform distribution
Answer: c) Exponential distribution

39. If two events A and B are not independent, what is true about their joint probability?

a) P(A and B) = P(A) * P(B)
b) P(A and B) = P(A) + P(B)
c) P(A and B) = P(A) * P(B|A)
d) P(A and B) = P(A) – P(B)
Answer: c) P(A and B) = P(A) * P(B|A)

40. In a Poisson distribution, what parameter represents the average rate of occurrences?

a) Mean
b) Variance
c) Rate
d) Standard deviation
Answer: c) Rate

Probability theory, Mcqs

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