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

## Chapter 3: Probability Distributions

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Q.1

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.1

#### MCQ [1 Mark]

Q.1 | Q 1

Choose the correct alternative:

The variance of a Binomial distribution is given by ______

• np

• pq

• npq

• sqrt("npq")

Q.1 | Q 2

Choose the correct alternative:

f(x) is c.d.f. of discete r.v. X whose distribution is

 xi – 2 – 1 0 1 2 pi 0.2 0.3 0.15 0.25 0.1

then F(– 3) = ______

• 0

• 1

• 0.2

• 0.15

Q.1 | Q 3

Choose the correct alternative:

X : is number obtained on upper most face when a fair die is thrown then E(X) = ______

• 3.0

• 3.5

• 4.0

• 4.5

Q.1 | Q 4

Choose the correct alternative:

If p.m.f. of r.v. X is given below.

 x 0 1 2 P(x) q2 2pq p2

then Var(x) = ______

• p2

• q2

• pq

• 2pq

Q.1 | Q 5

Choose the correct alternative:

The expected value of the sum of two numbers obtained when two fair dice are rolled is ______

• 5

• 6

• 7

• 8

Q.1 | Q 6

Choose the correct alternative:

If X ~ B(20, 1/10), then E(x) = ______

• 2

• 5

• 4

• 3

Q.1 | Q 7

Choose the correct alternative:

A sequence of dichotomous experiments is called a sequence of Bernoulli trials if it satisfies ______

• The trials are independent.

• The probability of success remains the same in all trials.

• The trials are independent but not the probability of success remains the same in all trials.

• both trials are independent but not the probability of success remains the same in all trials.

Q.1 | Q 8

Choose the correct alternative:

For the Poisson distribution ______

• Mean = E(X) = m

• Var(X) = m

• Mean = E(X) = m and Var(X) = m

• Mean = E(X) ≠ m and Var(X) = m

Q.1 | Q 9

Choose the correct alternative:

A distance random variable X is said to have the Poisson distribution with parameter m if its p.m.f. is given by P(x) = ("e"^(-"m")"m"^"x")/("x"!) the condition for m is ______

• m > 0

• m ≥ 0

• m ≠ 1

• m = 0

Q.2

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.2

#### Fill in the blanks [1 Mark]

Q.2 | Q 1

The values of discrete r.v. are generally obtained by ______

Q.2 | Q 2

The values of continuous r.v. are generally obtained by ______

Q.2 | Q 3

If X is discrete random variable takes the values x1, x2, x3, … xn, then sum_("i" = 1)^"n" "P"(x_"i") = ______

Q.2 | Q 4

E(x) is considered to be ______ of the probability distribution of x.

Q.2 | Q 5

In Binomial distribution, probability of success ______ from trial to trial

Q.2 | Q 6

In Binomial distribution if n is very large and probability of success of p is very small such that np = m (constant), then ______ distribution is applied

Q.2 | Q 7

When n is very large and p is very small in the binomial distribution, then X follows the Poission distribution with prameter m = ______

Q.3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.3

#### [1 Mark]

Q.3 | Q 1

State whether the following statement is True or False:

X is the number obtained on upper most face when a die is thrown, then E(x) = 3.5

• True

• False

Q.3 | Q 2

State whether the following statement is True or False:

If f(x) = {:("k"x  (1 - x)",", "for"  0 < x < 1),(= 0",", "otherwise"):}
is the p.d.f. of a r.v. X, then k = 12

• True

• False

Q.3 | Q 3

State whether the following statement is True or False:

If X ~ B(n, p) and n = 6 and P(X = 4) = P(X = 2), then p = 1/2

• True

• False

Q.3 | Q 4

State whether the following statement is True or False:

If a r.v. X assumes the values 1, 2, 3, …….., 9 with equal probabilities, then E(X) = 5

• True

• False

Q.3 | Q 5

State whether the following statement is True or False:

Let X ~ B(n, p), then the mean or expected value of r.v. X is denoted by E(X). It is also denoted by E(X) and is given by µ = E(X) = npq

• True

• False

Q.3 | Q 6

State whether the following statement is True or False:

A discrete random variable X is said to follow the Poisson distribution with parameter m ≥ 0 if its p.m.f. is given by P(X = x) = ("e"^(-"m")"m"^"x")/"x", x = 0, 1, 2, .....

• True

• False

Q.3 | Q 7

State whether the following statement is True or False:

For the Binomial distribution, Mean E(X) = m and Variance = Var(X) = m

• True

• False

Q.3 | Q 8

State whether the following statement is True or False:

If n is very large and p is very small then X follows Poisson distribution with n = mp

• True

• False

Q.3 | Q 9

State whether the following statement is True or False:

The cumulative distribution function (c.d.f.) of a continuous random variable X is denoted by F and defined by

F(x) = {:(0",",  "for all"  x ≤ "a"),( int_"a"^x  f(x) "d"x",",  "for all"  x ≥ "a"):}

• True

• False

Q.4

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.4

#### Solve the following [3 Marks]

Q.4 | Q 1. a)

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

Q.4 | Q 1. b)

Find the probability distribution of number of number of tails in three tosses of a coin

Q.4 | Q 1. c)

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

Q.4 | Q 2

A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.

Q.4 | Q 3

Find the expected value and variance X using the following p.m.f.

 x – 2 – 1 0 1 2 P(x) 0.2 0.3 0.1 0.15 0.25
Q.4 | Q 4

Find the mean of number of heads in three tosses of a fair coin

Q.4 | Q 5

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X

Q.4 | Q 6

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes

Q.4 | Q 7

Given that X ~ B(n, p), if n = 10 and p = 0.4, find E(X) and Var(X)

Q.4 | Q 8

If X has Poisson distribution with m = 1, then find P(X ≤ 1) given e−1 = 0.3678

Q.4 | Q 9

If X has Poisson distribution with parameter m and P(X = 2) = P(X = 3), then find P(X ≥ 2). Use e−3 = 0.0497

Q.5

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.5

#### Solve the following [4 Marks]

Q.5 | Q 1. (i)

Solve the following problem :

A random variable X has the following probability distribution.

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

Determine k

Q.5 | Q 1. (ii)

Solve the following problem :

A random variable X has the following probability distribution.

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

Determine P(X < 3)

Q.5 | Q 1. (iii)

Solve the following problem:

A random variable X has the following probability distribution.

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

Determine P(0 < X < 3)

Q.5 | Q 1. (iv)

A random variable X has the following probability distribution:

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

Determine P(X > 4)

Q.5 | Q 2. (i)

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8),  0 < x < 2),(0,   "otherwise".):}
Determine the c.d.f. of X and hence find P(X < 1)

Q.5 | Q 2. (ii)

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}
Determine the c.d.f. of X and hence find P(X < –2)

Q.5 | Q 2. (iii)

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8),  0 < x < 2),(0, "otherwise".):}
Determine the c.d.f. of X and hence find P(X > 0)

Q.5 | Q 2. (iv)

The p.d.f. of a continuous r.v. X is

f(x) = {((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}
Determine the c.d.f. of X and hence find P(1 < X < 2)

Q.5 | Q 3

If a r.v. X has p.d.f f(x) = {("c"/x","  1 < x < 3"," "c" > 0),(0","  "otherwise"):}
Find c, E(X), and Var(X). Also Find F(x).

Q.5 | Q 4. i)

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes

Q.5 | Q 4. ii)

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes

Q.5 | Q 4. (iii)

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at most 2 successes

Q.5 | Q 5

The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

Q.5 | Q 6

The number of complaints which a bank manager receives per day follows a Poisson distribution with parameter m = 4. Find the probability that the manager receives a) only two complaints on a given day, b) at most two complaints on a given day. Use e−4 = 0.0183.

Q.5 | Q 7

Defects on plywood sheet occur at random with the average of one defect per 50 sq. ft. Find the probability that such a sheet has (i) no defect, (ii) at least one defect. Use e−1 = 0.3678.

Q.5 | Q 8. (i)

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has exactly 5 rats inclusive. Given e-5  = 0.0067.

Q.5 | Q 8. (ii)

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has more than 5 rats inclusive. Given e-5  = 0.0067.

Q.5 | Q 8. (iii)

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has between 5 and 7 rats inclusive. Given e−5 = 0.0067.

Q.6

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 3 Probability Distributions Q.6

#### Activities [4 Marks]

Q.6 | Q 1. i)

The probability distribution of a discrete r.v.X is as follows.

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

Complete the following activity.

Solution: Since sum"p"_"i" = 1

k = square

Q.6 | Q 1. ii)

The probability distribution of a discrete r.v.X is as follows.

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

Complete the following activity.

Solution: Since sum"p"_"i" = 1

P(X ≤ 4) = square + square + square + square = square

Q.6 | Q 1. (iii)

The probability distribution of a discrete r.v.X is as follows.

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

Complete the following activity.

Solution: Since sum"p"_"i" = 1

P(X ≥ 3) = square - square - square  = square

Q.6 | Q 2

Using the following activity, find the expected value and variance of the r.v.X if its probability distribution is as follows.

 x 1 2 3 P(X = x) 1/5 2/5 2/5

Solution: µ = E(X) = sum_("i" = 1)^3 x_"i""p"_"i"

E(X) = square + square + square = square

Var(X) = "E"("X"^2) - {"E"("X")}^2

= sum"X"_"i"^2"P"_"i" - [sum"X"_"i""P"_"i"]^2

= square - square

= square

Q.6 | Q 3

Let X ~ B(n, p). If n = 10 and E(X) = 5, using the following activity find p and Var(X)

Solution: E(X) = square = 5 square "p" = square, "q" = square

Var(X) = square

Q.6 | Q 4

The probability that a bomb will hit the target is 0.8. Using the following activity, find the probability that, out of 5 bombs, exactly 2 will miss the target

Solution: Let p = probability that bomb miss the target

∴ q = square, p = square, n = 5.

X ~ B(5, square), P(x) = ""^"n""C"_x"P"^x"q"^("n" - x)

P(X = 2) =  ""^5"C"_2  square = square

Q.6 | Q 5

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, using the following activity find the value of m.

Solution: X : Follows Poisson distribution

∴ P(X) = ("e"^-"m" "m"^x)/(x!), P(X = 1) = 0.4 and P(X = 2) = 0.2

∴ P(X = 1) = square P(X = 2).

("e"^-"m" "m"^x)/(1!) = square ("e"^-"m" "m"^2)/(2!),

"e"^-"m" = square  "e"^-"m" "m"/2, m ≠ 0

∴ m = square

## Chapter 3: Probability Distributions

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

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 3 Probability Distributions are Mean of a Random Variable, Types of Random Variables, Random Variables and Its Probability Distributions, Probability Distribution of Discrete Random Variables, Probability Distribution of a Continuous Random Variable, Binomial Distribution, Bernoulli Trial, Mean of Binomial Distribution (P.M.F.), Variance of Binomial Distribution (P.M.F.), Poisson Distribution.

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