# Probability and Statistics MCQ Questions with answer

Probability and Statistics MCQ Questions with answer keys for preparation of academic and competitive exams of various institutions in 2024

Q1. If P(A) = 0.35, P(B) = 0.73, P(A∩B) = 0.14, find P(A U B)

Ans. 0.86

Q2. When two dice are tossed, what is the probability of getting four as the sum of the face numbers?

Ans. 1/12

Q3: If A and B are independent and P (A) = 1/3 P(B) = 1/4, find P(A∩B)

Ans. 1/12

Q4. A coin is tossed thrice. What is the chance of getting all heads?

Ans. 1/8

Q5: If P(A) = 0.9, P(B/A) = 0.8, find P(A ÇB)

Ans. 0.72

Q6: If P(A) = 1/3, P(B) = 3/4 , find (P È B) = 11/12, find P(A/B) and P(B/A)

Ans. 2/9, 1/2

Q7: There are 3 true coins and 1 false coin with ‘head’ on both sides. A coin is chosen at random and tossed 4 times. If ‘Head’ occurs all the 4 times. If the head occurs at all the 4 times, what is the probability that the false coin has chosen and used?

Ans. 16/19

Q8: A random variable X has the following probability function

x | 0 | 1 | 2 | 3 | 4 |

P(X=x) | c | 3c | 5c | 7c | 9c |

i) Find the value of c.(ii) Find P(X<4), P(X>4), P(0<X<4)

ii) Find the distribution function of X.

Ans. c = 1/25 and 16/25, 9/25, 3/5

Q9: The joint distribution of X and Y is given by

F(x,y) = x+y / 21, x = 1,2,3; y = 1,2 Find the marginal distributions.

Ans. 5/21, 7/21, 9/21, and 9/21, 12/21,

Q10: Two coins are thrown simultaneously and let X denotes the number of heads. Calculate all possible outcomes and find the probability distribution of X. Also calculate the expected values of X.

Ans. Let H denotes the head and T denotes the tail. So all possible outcomes after tossing two coins simultaneously are

Outcomes Number of heads Probability of occurrence

HH 2 2/4 = ½

HT 1 1/4

TH 1 1/4

TT 0 0

So, E(X) = 0.f(x) + 1.f(x) + 2.f(x) = 1. 2/4 + 2. ½ = 3/2

Q11: Let pmf of discrete random variable is p(x) = 1/n, where x = 1,2,3,4,….. Find E(X).

Ans. (n+1)/2

Q12: By throwing a fair die, a game is played. A player wins Rs. 20 if 2 turns up, Rs. 40 if 4 turns up and loses Rs.30 if 6 turns up. And he never loses or gains if any other number turns up. Find the expected value of money won.

Ans. E(X)=5

Q13: X is a random variable whose density function is

Q14: Find the mean of the random variable X, if its pdf is given by

f(x)= 6x(1-x), 0≤x≤1.

Ans. E(X)=1/2

Q15: If the joint pdf of (X,Y) is given by f(x,y)= 24y(1-x), 0≤y≤x≤1, find E(XY).

Ans. E(XY)=4/15

Q16: Assume two random variables X and Y jointly distributed with density function f(x,y)= x+y¸ 0≤x≤1, 0≤y≤1. Find Cov(X,Y).

Ans. -1/144.

Q17. Two events A and B are independent if and only if P(AUB) is equal to which among the following?

a. P(A) U P(B)

**b. P(A)*P(B)
**c. P(A) ∩ P(B)

d. P(A ∩ B)

Q18. Two or more events are independent if the occurrence of one ____ the occurrence of the other.

a. is the same as the

b. depends on

**c. does not affect the
**d. differs from the

Q19. All possible outcomes of an experiment are called which among the following?

a. mutually independent events

b. mutually exclusive events

c. independent events

**d. exhaustive events**

Q20. If f is a function (mapping) from S into the set R of all real numbers and X = f(s), sϵS, then which among the following holds?

a. X is an independent variable

**b. X is called a random variable on S
**c. S is called a random variable on S

d. S is an independent variable

Q21. Which is not a possible outcome of tossing 3 coins together?

a. HHTT

b. HTHH

**c. TTTT
**d. All of these

Q22. If (X, Y) is a two-dimensional random variable, and if X & Y are independent then V(X+Y)

**a. V(X) + V(Y)
**b. V(X) – V(Y)

c. V(X)*V(Y)

d. [V(X)+V(Y)] / sqrt[{V(X)*V(Y)}]

Q23. The measure of association between two random variables is called which of the following?

a. convergence

**b. covariance
**c. conditional variance

d. conditional expectation

Q24. If X and Y are independent random variables, then E(XY) is equal to which of the following?

a. E(X)+E(Y)

**b. E(X)*E(Y)
**c. [E(X)+E(Y)] / sqrt[{E(X)*E(Y)}]

d. [E(X)-E(Y)] / sqrt[{E(X)*E(Y)}]

Q25. The moment generating function of the sum of a number of the independent random variable is ____ the product of their respective moment generating functions.

a. greater than or equal to

b. lesser than

**c. equal to
**d. lesser than or equal to

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