#### Chapters

Chapter 2: Complex Numbers

Chapter 3: Theory of Equations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Two Dimensional Analytical Geometry-II

Chapter 6: Applications of Vector Algebra

Chapter 7: Applications of Differential Calculus

Chapter 8: Differentials and Partial Derivatives

Chapter 9: Applications of Integration

Chapter 10: Ordinary Differential Equations

Chapter 11: Probability Distributions

Chapter 12: Discrete Mathematics

## Chapter 11: Probability Distributions

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Probability DistributionsExercise 11.1 [Page 184]

Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its inverse images

In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images

An urn contains 5 mangoes and 4 apples. Three fruits are taken at random. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images

Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win ₹ 15 for each red ball selected and we lose ₹ 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images

A six sided die is marked ‘2’ on one face, ‘3’ on two of its faces, and ‘4’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Probability DistributionsExercise 11.2 [Page 194]

Three fair coins are tossed simultaneously. Find the probability mass function for a number of heads that occurred

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the probability mass function

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the cumulative distribution function

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find P(4 ≤ X < 10)

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find P(X ≥ 6)

Find the probability mass function and cumulative distribution function of a number of girl children in families with 4 children, assuming equal probabilities for boys and girls

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by

`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`

Find the value of k

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by

`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`

Find cumulative distribution function

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by

`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`

Find P(X ≥ 1)

The cumulative distribution function of a discrete random variable is given by

F(x) = `{{:(0, - oo < x < - 1),(0.15, - 1 ≤ x < 0),(0.35, 0 ≤ x < 1),(0.60, 1 ≤ x < 2),(0.85, 2 ≤ x < 3),(1, 3 ≤ x < oo):}`

Find the probability mass function

The cumulative distribution function of a discrete random variable is given by

F(x) = `{{:(0, - oo < x < - 1),(0.15, - 1 ≤ x < 0),(0.35, 0 ≤ x < 1),(0.60, 1 ≤ x < 2),(0.85, 2 ≤ x < 3),(1, 3 ≤ x < oo):}`

Find P(X < 1)

The cumulative distribution function of a discrete random variable is given by

F(x) = `{{:(0, - oo < x < - 1),(0.15, - 1 ≤ x < 0),(0.35, 0 ≤ x < 1),(0.60, 1 ≤ x < 2),(0.85, 2 ≤ x < 3),(1, 3 ≤ x < oo):}`

Find P(X ≥ 2)

A random variable X has the following probability mass function.

x |
1 | 2 | 3 | 4 | 5 |

F(x) |
k^{2} |
2k^{2} |
3k^{2} |
2k | 3k |

Find the value of k

A random variable X has the following probability mass function.

x |
1 | 2 | 3 | 4 | 5 |

F(x) |
k^{2} |
2k^{2} |
3k^{2} |
2k | 3k |

Find P(2 ≤ X < 5)

A random variable X has the following probability mass function.

x |
1 | 2 | 3 | 4 | 5 |

F(x) |
k^{2} |
2k^{2} |
3k^{2} |
2k | 3k |

Find P(X > 3)

The cumulative distribution function of a discrete random variable is given by

F(x) = `{{:(0, "for" - oo < x < 0),(1/2, "for" 0 ≤ x < 1),(3/5, "for" 1 ≤ x < 2),(4/5, "for" 2 ≤ x < 4),(9/5, "for" 3 ≤ x < 4),(1, "for" ≤ x < oo):}`

Find the probability mass function

The cumulative distribution function of a discrete random variable is given by

F(x) = `{{:(0, "for" - oo < x < 0),(1/2, "for" 0 ≤ x < 1),(3/5, "for" 1 ≤ x < 2),(4/5, "for" 2 ≤ x < 4),(9/5, "for" 3 ≤ x < 4),(1, "for" ≤ x < oo):}`

Find P(X < 3)

The cumulative distribution function of a discrete random variable is given by

F(x) = `{{:(0, "for" - oo < x < 0),(1/2, "for" 0 ≤ x < 1),(3/5, "for" 1 ≤ x < 2),(4/5, "for" 2 ≤ x < 4),(9/5, "for" 3 ≤ x < 4),(1, "for" ≤ x < oo):}`

Find P(X ≥ 2)

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Probability DistributionsExercise 11.3 [Pages 202 - 203]

The probability density function of X is given by

`f(x) = {{:(kx"e"^(-2x), "for" x > 0),(0, "for" x ≤ 0):}`

Find the value of k

The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`

Find P(0.2 ≤ X < 0.6)

The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`

Find P(1.2 ≤ X < 1.8)

The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`

Find P(0.5 ≤ X < 1.5)

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function

`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`

Find the value of k

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function

`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`

Find the distribution function

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function

`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`

Find the probability that daily sales will fall between 300 litres and 500 litres?

The probability density function of X is given by

`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`

Find the value of k

The probability density function of X is given by

`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`

Find the distribution function

The probability density function of X is given by

`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`

Find P(X < 3)

The probability density function of X is given by

`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`

Find P(5 ≤ X)

The probability density function of X is given by

`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`

Find P(X ≤ 4)

If X is the random variable with probability density function f(x) given by,

`f(x) = {{:(x + 1",", -1 ≤ x < 0),(-x +1",", 0 ≤ x < 1),(0, "otherwise"):}`

then find the distribution function F(x)

If X is the random variable with probability density function f(x) given by,

`f(x) = {{:(x + 1",", -1 ≤ x < 0),(-x +1",", 0 ≤ x < 1),(0, "otherwise"):}`

then find P(– 0.5 ≤ x ≤ 0.5)

If X is the random variable with distribution function F(x) given by,

F(x) = `{{:(0",", - oo < x < 0),(1/2(x^2 + x)",", 0 ≤ x ≤ 1),(1",", 1 ≤ x < oo):}`

then find the probability density function f(x)

If X is the random variable with distribution function F(x) given by,

F(x) = `{{:(0",", - oo < x < 0),(1/2(x^2 + x)",", 0 ≤ x ≤ 1),(1",", 1 ≤ x < oo):}`

then find P(0.3 ≤ X ≤ 0.6)

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Probability DistributionsExercise 11.4 [Page 210]

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:(1/10, x = 2"," 5),(1/5, x = 0"," 1"," 3"," 4):}`

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:((4 - x)/6, x = 1"," 2"," 3),(0, "otherwise"):}`

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:(2(x - 1), 1 < x ≤ 2),(0, "otherwise"):}`

`f(x) = {{:(1/2 "e"^(x/2), "for" x > 0),(0, "otherwise"):}`

Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X

If µ and σ^{2} are the mean and variance of the discrete random variable X and E(X + 3) = 10 and E(X + 3)^{2} = 116, find µ and σ^{2 }

Four fair coins are tossed once. Find the probability mass function, mean and variance for a number of heads that occurred

A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station. Let X denote the amount of time, in minutes, that the student waits for the train from the time he reaches the train station. It is known that the pdf of X is

`f(x) = {{:(1/30, 0 < x < 30),(0, "elsewhere"):}`

Obtain and interpret the expected value of the random variable X

The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function

`f(x) = {{:(3"e"^(-3x), x > 0),(0, "eleswhere"):}`

Find the expected life of this electronic equipment

The probability density function of the random variable X is given by

`f(x) = {{:(16x"e"^(-4x), x > 0),(0, x ≤ 0):}`

find the mean and variance of X

A lottery with 600 tickets gives one prize of ₹ 200, four prizes of ₹ 100, and six prizes of ₹ 50. If the ticket costs is ₹ 2, find the expected winning amount of a ticket

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Probability DistributionsExercise 11.5 [Page 218]

Compute P(X = k) for the binomial distribution, B(n, p) where

n = 6, p = `1/3`, k = 3

Compute P(X = k) for the binomial distribution, B(n, p) where

n = 10, p = `1/5`, k = 4

Compute P(X = k) for the binomial distribution, B(n, p) where

n = 9, p = `1/2`, k = 7

The probability that Mr.Q hits a target at any trial is `1/4`. Suppose he tries at the target 10 times. Find the probability that he hits the target exactly 4 times

The probability that Mr.Q hits a target at any trial is `1/4`. Suppose he tries at the target 10 times. Find the probability that he hits the target at least one time

Using binomial distribution find the mean and variance of X for the following experiments.

A fair coin is tossed 100 times, and X denote the number of heads

Using binomial distribution find the mean and variance of X for the following experiments.

A fair die is tossed 240 times and X denotes the number of times that four appeared

The probability that a certain kind of component will survive a electrical test is `3/4`. Find the probability that exactly 3 of the 5 components tested survive

A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be at least one defective item

A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be exactly two defective items?

If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights exactly 10 will have a useful life of at least 600 hours

If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights at least 11 will have a useful life of at least 600 hours

If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights at least 2 will not have a useful life of at least 600 hours

The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find the probability mass function

The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find P(X = 3)

The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find P(X ≥ 2)

If X ~ B(n, p) such that 4P(X = 4) = P(x = 2) and n = 6. Find the distribution, mean and standard deviation of X

In a binomial distribution consisting of 5 independent trials, the probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. Find the mean and variance of the random variable

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Probability DistributionsExercise 11.6 [Pages 218 - 221]

#### MCQ

Choose the correct alternative:

Let X be random variable with probability density function

`f(x) = {(2/x^3, x ≥ 1),(0, x < 1):}`

Which of the following statement is correct?

Both mean and variance exist

Mean exists but variance does not exist

Both mean and variance do not exist

Variance exists but mean does not exist.

Choose the correct alternative:

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is

`f(x) = {{:(1/l, 0 < x < l),(0, l ≤ x < 2l):}`

The mean and variance of the shorter of the two pieces are respectively

`l/2, l^2/3`

`l/2, l^2/6`

`l, l^2/12`

`l/2, l^2/12`

Choose the correct alternative:

Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins ₹ 36, otherwise he loses ₹ k^{2}, where k is the face that comes up k = {1, 2, 3, 4, 5}. The expected amount to win at this game in ₹ is

`19/6`

`- 19/6`

`3/2`

`- 3/2`

Choose the correct alternative:

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

1

2

3

4

Choose the correct alternative:

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

6

4

3

2

Choose the correct alternative:

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

i + 2n, i = 0, 1, 2 …, n

2i – n, i = 0, 1, 2 …, n

n – i, i = 0, 1, 2 …, n

2i + 2n, i = 0, 1, 2 …, n

Choose the correct alternative:

If the function f(x) = `1/12` for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?

0 and 12

5 and 17

7 and 19

16 and 24

Choose the correct alternative:

Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are

50, 40

40, 50

40, 75, 40

41, 41

Choose the correct alternative:

Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with Probability 0.5. Assume that the results of the flips are independent and let X equal the total number of heads that result. The value of E[X] is

0 11

1.1

11

1

Choose the correct alternative:

On a multiple-choice exam with 3 possible destructive for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

`11/243`

`3/8`

`1/243`

`5/243`

Choose the correct alternative:

If P(X = 0) = 1 – P(X = 1). If E[X] = 3 Var(X), then P(X = 0) is

`2/3`

`2/5`

`1/5`

`1/3`

Choose the correct alternative:

If X is a binomial random variable with I expected value 6 and variance 2.4, Then P(X = 5) is

`(10/5)(3/5)^6(2/5)^4`

`(10/5)(3/5)^10`

`(10/5)(3/5)^4(2/5)^4`

`(10/5)(3/5)^5(2/5)^5`

Choose the correct alternative:

The random variable X has the probability density function

`f(x) = {{:("a"x + "b", 0 < x < 1),(0, "otherwise"):}`

and E(X) = `7/12`, then a and b are respectively

1 and `1/2`

`1/2` and 1

2 and 1

1 and 2

Choose the correct alternative:

Suppose that X takes on one of the values 0, 1 and 2. If for some constant k, P(X = i) = kP(X = i – 1) for i = 1, 2 and P(X = 0) = `1/7`. Then the value of k is

1

2

3

4

Choose the correct alternative:

Which of the following is a discrete random variable?

I. The number of cars crossing a particular signal in a day.

II. The number of customers in a queue to buy train tickets at a moment.

III. The time taken to complete a telephone call.

I and II

II only

III only

II and III

Choose the correct alternative:

If `f(x) = {{:(2x, 0 ≤ x ≤ "a"),(0, "otherwise"):}` is a probability density function of a random variable, then the value of a is

1

2

3

4

Choose the correct alternative:

The probability mass function of a random variable is defined as:

x |
– 2 | – 1 | 0 | 1 | 2 |

f(x) |
k | 2k | 3k | 4k | 5k |

Then E(X ) is equal to:

`1/15`

`1/10`

`1/3`

`2/3`

Choose the correct alternative:

Let X have a Bernoulli distribution with a mean of 0.4, then the variance of (2X – 3) is

0.24

0.24

0.6

0.96

Choose the correct alternative:

If in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the probability of success is

0.125

0.25

0.375

0.75

Choose the correct alternative:

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

`57/20^3`

`57/20^2`

`19^3/20^3`

`57/20`

## Chapter 11: Probability Distributions

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 11 - Probability Distributions

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 11 (Probability Distributions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 12th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 11 Probability Distributions are Introduction to Probability Distributions, Random Variable, Types of Random Variables, Continuous Distributions, Mathematical Expectation, Theoretical Distributions: Some Special Discrete Distributions.

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