SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 3 - Probability Distributions [Latest edition]

Chapter 3: Probability Distributions

MCQ

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 3 Probability Distributions MCQ

2 Mark each

MCQ | Q 1

Let the p.m.f. of a random variable X be P(x) = (3 - x)/10, for x = −1, 0, 1, 2 = 0, otherwise Then E(x) is ______

• 1

• 2

• 0

• –1

MCQ | Q 2

c.d.f. of a discrete random variable X is

• an identity function

• a step function

• an even function

• an odd function

MCQ | Q 3

If X denotes the number on the uppermost face of cubic die when it is tossed, then E(X) is ______

• 2/7

• 7/2

• 1

• 1/2

MCQ | Q 4

A random variable X has the following probability distribution

 X 2 3 4 P(x) 0.3 0.4 0.3

Then the variance of this distribution is

• 0.6

• 0.7

• 0.77

• 0.66

MCQ | Q 5

For the random variable X, if V(X) = 4, E(X) = 3, then E(x2) is ______

• 9

• 13

• 12

• 7

MCQ | Q 6

If a d.r.v. X takes values 0, 1, 2, 3, … with probability P(X = x) = k(x + 1) × 5–x, where k is a constant, then P(X = 0) = ______

• 7/25

• 16/25

• 18/25

• 19/25

MCQ | Q 7

The p.m.f. of a d.r.v. X is P(X = x) = {{:(((5),(x))/2^5",", "for"  x = 0","  1","  2","  3","  4","  5),(0",", "otherwise"):} If a = P(X ≤ 2) and b = P(X ≥ 3), then

• a < b

• a > b

• a = b

• a + b = 2

MCQ | Q 8

If the p.m.f. of a d.r.v. X is P(X = x) = {{:(x/("n"("n" + 1))",", "for"  x = 1","  2","  3","  .... "," "n"),(0",", "otherwise"):}, then E(X) = ______

• "n" + 1/2

• "n"/3 + 1/6

• "n"/2 + 1/5

• "n" + 1/3

MCQ | Q 9

If the p.m.f. of a d.r.v. X is P(X = x) = {{:(("c")/x^3",", "for"  x = 1","  2","  3","),(0",", "otherwise"):} then E(X) = ______

• 343/297

• 294/251

• 297/294

• 294/297

MCQ | Q 10

If a d.r.v. X has the following probability distribution:

 X –2 –1 0 1 2 3 P(X = x) 0.1 k 0.2 2k 0.3 k

then P(X = –1) is ______

• 1/10

• 2/10

• 3/10

• 4/10

MCQ | Q 11

If a d.r.v. X has the following probability distribution:

 X 1 2 3 4 5 6 7 P(X = x) k 2k 2k 3k k2 2k2 7k2 + k

then k = ______

• 1/7

• 1/8

• 1/9

• 1/10

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 3 Probability Distributions Very Short Answers

1 Mark

Very Short Answers | Q 1

Let X represent the difference between number of heads and number of tails obtained when a coin is tossed 6 times. What are possible values of X?

Very Short Answers | Q 2

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?

Very Short Answers | Q 3. (i)

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 1 2 P(X) 0.4 0.4 0.2
Very Short Answers | Q 3. (ii)

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 1 2 3 4 P(X) 0.1 0.5 0.2 − 0.1 0.2
Very Short Answers | Q 3. (iii)

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 1 2 P(X) 0.1 0.6 0.3
Very Short Answers | Q 3. (iv)

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 Y −1 0 1 P(Y) 0.6 0.1 0.2
Very Short Answers | Q 4

Find mean for the following probability distribution.

 X 0 1 2 3 P(X = x) 1/6 1/3 1/3 1/6
Very Short Answers | Q 5

State if the following is not the probability mass function of a random variable. Give reasons for your answer

 X 3 2 1 0 −1 P(X = x) 0.3 0.2 0.4 0 0.05

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 3 Probability Distributions Short Answers I

2 Marks

Short Answers I | Q 1

Find the expected value and variance of r.v. X whose p.m.f. is given below.

 X 1 2 3 P(X = x) 1/5 2/5 2/5
Short Answers I | Q 2

Find the probability distribution of number of heads in two tosses of a coin

Short Answers I | Q 3

The probability distribution of X is as follows:

 X 0 1 2 3 4 P(X = x) 0.1 k 2k 2k k

Find k and P[X < 2]

Short Answers I | Q 4. (i)

Solve the following problem :

Following is the probability distribution of a r.v.X.

 X – 3 – 2 –1 0 1 2 3 P(X = x) 0.05 0.1 0.15 0.2 0.25 0.15 0.1

Find the probability that X is positive.

Short Answers I | Q 4. (ii)

Solve the following problem:

Following is the probability distribution of a r.v.X.

 X – 3 – 2 –1 0 1 2 3 P(X = x) 0.05 0.1 0.15 0.2 0.25 0.15 0.1

Find the probability that X is odd.

Short Answers I | Q 5

In the p.m.f. of r.v. X

 X 1 2 3 4 5 P (X) 1/20 3/20 a 2a 1/20

Find a and obtain c.d.f. of X.

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 3 Probability Distributions Short Answers II

3 Marks

Short Answers II | Q 1

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 appears on at least one die

Short Answers II | Q 2

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Short Answers II | Q 3

A random variable X has the following probability distribution :

 X 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine :
(i) k
(ii) P(X < 3)
(iii) P( X > 4)

Short Answers II | Q 4

Find the probability distribution of the number of doublets in three throws of a pair of dice

Short Answers II | Q 5

Find the mean and variance of the number randomly selected from 1 to 15

Short Answers II | Q 6

Let the p.m.f. of r.v. X be P(x)  {{:(((3 - x)/10",", "for"  x = -1",", 0",", 1",", 2)),((0",", "otherwise")):} Calculate E(X) and Var(X)

Short Answers II | Q 7

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as six appears on at least one die

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 3 Probability Distributions Long Answers III

4 Marks

Long Answers III | Q 1

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X

Long Answers III | Q 2

Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings drawn

Long Answers III | Q 3

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).

Long Answers III | Q 4

In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var(X).

Long Answers III | Q 5

The following is the c.d.f. of r.v. X:

 X −3 −2 −1 0 1 2 3 4 F(X) 0.1 0.3 0.5 0.65 0.75 0.85 0.9 1

Find p.m.f. of X.
i. P(–1 ≤ X ≤ 2)
ii. P(X ≤ 3 / X > 0).

Long Answers III | Q 6

Solve the following problem :

A player tosses two coins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears, and ₹ 2 if no head appears. Find the expected value and variance of winning amount.

Long Answers III | Q 7

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Long Answers III | Q 8

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of X.

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 3 - Probability Distributions

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 3 Probability Distributions are Random Variables and Its Probability Distributions, Types of Random Variables, Probability Distribution of Discrete Random Variables, Probability Distribution of a Continuous Random Variable, Variance of a Random Variable.

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