**NCERT solutions for exercise 13.4 class 12 maths chapter 13 **you will learn some important concepts like random variable, the probability distribution of random variable, calculating mean and variance of a probability distribution. . The concept of the random variable is a very important concept in statistics. You are advised to be thorough with **class 12 maths ch 13 ex 13.4** as this is very frequently asked in the board exams. There are 17 questions in** class 12th maths chapter 13 exercise 13.4** which you should try to solve on your own. You can take help from these **exercise 13.4 class 12 maths** solutions if you are not able to solve them on your own. These solutions are prepared in a detailed manner which could be understood very easily. You can also check for NCERT solutions.

**Also, see**

- NCERT solutions for class 12 maths chapter 13 Probability Exercise 13.2
- NCERT solutions for class 12 maths chapter 13 Probability Exercise 13.3
- NCERT solutions for class 12 maths chapter 13 Probability Exercise 13.5
- NCERT solutions for class 12 maths chapter 13 Probability Miscellaneous Exercise

**Question:1(i) **State which the following are not the probability distributions of a random variable. Give reasons for your answer.

**Answer:**

As we know the sum of probabilities of a probability distribution is 1.

Sum of probabilities

The given table is the probability distributions of a random variable.

**Question:1(ii) ** State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

**Answer:**

As we know probabilities cannot be negative for a probability distribution .

The given table is not a the probability distributions of a random variable.

**Question:1(iii) ** State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

**Answer:**

As we know sum of probabilities of a probability distribution is 1.

Sum of probablities

The given table is not a the probability distributions of a random variable because sum of probabilities is not 1.

**Question:1(iv) **State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

**Answer:**

As we know sum of probabilities of a probability distribution is 1.

Sum of probablities

The given table is not a the probability distributions of a random variable because sum of probabilities is not 1.

**Question:2** An urn contains red and black balls. Two balls are randomly drawn. Let represent the number of black balls. What are the possible values of Is a random variable ?

**Answer:**

B = black balls

R = red balls

The two balls can be selected as BR,BB,RB,RR.

X = number of black balls.

Hence, possible values of X can be 0, 1 and 2.

Yes, X is a random variable.

**Question:3** Let represent the difference between the number of heads and the number of tails obtained when a coin is tossed times. What are possibl valuess of ?

**Answer:**

The difference between the number of heads and the number of tails obtained when a coin is tossed times are :

Thus, possible values of X are 0, 2, 4 and 6.

**Question:4(i)** Find the probability distribution of

number of heads in two tosses of a coin.

**Answer:**

When coin is tossed twice then sample space

Let X be number of heads.

X can take values of 0,1,2.

Table is as shown :

X | 0 | 1 | 2 |

P(X) |

**Question:4(ii)** Find the probability distribution of

number of tails in the simultaneous tosses of three coins.

**Answer:**

When 3 coins are simultaneous tossed then sample space

Let X be number of tails.

X can be 0,1,2,3

X can take values of 0,1,2.

Table is as shown :

X | 0 | 1 | 2 | 3 |

P(X) |

**Question:4(iii) **Find the probability distribution of

number of heads in four tosses of a coin.

**Answer:**

When coin is tossed 4 times then sample space

Let X be number of heads.

X can be 0,1,2,3,4

Table is as shown :

X | 0 | 1 | 2 | 3 | 4 |

P(X) |

**Question:5(i)** 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

**Answer:**

When a die is tossed twice , total outcomes = 36

Number less than or equal to 4 in both toss :

Number less than or equal to 4 in first toss and number more than or equal to 4 in second toss + Number less than or equal to 4 in second toss and number more than or equal to 4 in first toss:

Number less than 4 in both tosses :

Probability distribution is as :

X | 0 | 1 | 2 |

P(X) |

**Question:5(ii)** 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.

**Answer:**

When a die is tossed twice , total outcomes = 36

Six does not appear on any of the die :

Six appear on atleast one die :

Probability distribution is as :

X | 0 | 1 |

P(X) |

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

**Answer:**

Total bulbs = 30

defective bulbs = 6

Non defective bulbs

bulbs is drawn at random with replacement.

Let X : number of defective bulbs

4 Non defective bulbs and 0 defective bulbs :

3 Non defective bulbs and 1 defective bulbs :

2 Non defective bulbs and 2 defective bulbs :

1 Non defective bulbs and 3 defective bulbs :

0 Non defective bulbs and 4 defective bulbs :

the probability distribution of the number of defective bulbs is as :

X | 0 | 1 | 2 | 3 | 4 |

P(X) |

**Question:7 **A coin is biased so that the head is times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

**Answer:**

the coin is tossed twice, total outcomes =4

probability of getting a tail be x.

i.e.

Then

and

Let X : number of tails

No tail :

1 tail :

2 tail :

the probability distribution of number of tails are

X | 0 | 1 | 2 |

P(X) |

**Question:8(i)** A random variable X has the following probability distribution:

**Answer:**

Sum of probabilities of probability distribution of random variable is 1.

**Question:8(ii)** A random variable has the following probability distribution:

**Answer:**

**Question:8(iii)** A random variable has the following probability distribution:

**Answer:**

**Question:8(iv)** A random variable X has the following probability distribution:

**Answer:**

**Question:9(a)** The random variable X has a probability distribution P(X) of the following form, where k is some number :

Determine the value of

**Answer:**

Sum of probabilities of probability distribution of random variable is 1.

**Question:9(b)** The random variable has a probability distribution of the following form, where k is some number :

Find

**Answer:**

**Question:10** Find the mean number of heads in three tosses of a fair coin.

**Answer:**

Let X be the success of getting head.

When 3 coins are tossed then sample space

X can be 0,1,2,3

The probability distribution is as

X | 0 | 1 | 2 | 3 |

P(X) |

mean number of heads :

**Question:11** Two dice are thrown simultaneously. If denotes the number of sixes, find the expectation of .

**Answer:**

denotes the number of sixes, when two dice are thrown simultaneously.

X can be 0,1,2.

Not getting six on dice

Getting six on one time when thrown twice :

Getting six on both dice :

X | 0 | 1 | 2 |

P(X) |

Expectation of X = E(X)

**Question:12 **Two numbers are selected at random (without replacement) from the first six positive integers. Let denote the larger of the two numbers obtained. Find

**Answer:**

Two numbers are selected at random (without replacement) from the first six positive integers in ways.

denote the larger of the two numbers obtained.

X can be 2,3,4,5,6.

X=2, obsevations :

X=3, obsevations :

X=4, obsevations :

X=5, obsevations :

X=6, obsevations :

Probability distribution is as follows:

X | 2 | 3 | 4 | 5 | 6 |

P(X) |

**Question:13** Let denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of.

**Answer:**

denote the sum of the numbers obtained when two fair dice are rolled.

Total observations = 36

X can be 2,3,4,5,6,7,8,9,10,11,12

Probability distribution is as follows :

X | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

P(X) |

Standard deviation =

**Question:14** A class has students whose ages are and years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable Find mean, variance and standard deviation of .

**Answer:**

Total students = 15

probability of selecting a student :

The information given can be represented as frequency table :

X | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

f | 2 | 1 | 2 | 3 | 1 | 2 | 3 | 1 |

Probability distribution is as :

X | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

P(X) |

**Question:15** In a meeting, of the members favour and oppose a certain proposal. A member is selected at random and we take . if he opposed, and if he is in favour. Find and Var .

**Answer:**

Given :

Probability distribution is as :

X | 0 | 1 |

P(X) | 0.3 | 0.7 |

**Question:16** The mean of the numbers obtained on throwing a die having written 1 on three faces, on two faces and on one face is, Choose the correct answer in the following:

(A)

(B)

(C)

(D)

**Answer:**

X is number representing on die.

Total observations = 6

X | 1 | 2 | 5 |

P(X) |

Option B is correct.

**Question:17** Suppose that two cards are drawn at random from a deck of cards. Let be the number of aces obtained. Then the value of is Choose the correct answer in the following:

(A)

(B)

(C)

(D)

**Answer:**

X be number od aces obtained.

X can be 0,1,2

There 52 cards and 4 aces, 48 are non-ace cards.

The probability distribution is as :

X | 0 | 1 | 2 |

P(X) |

Option D is correct.

**More about ****NCERT Solutions for Class 12 Maths Chapter 13 Exercise 13.4:-**

A new concept called Random Variables and its Probability Distributions is introduced in **NCERT Solutions for Class 12 Maths Chapter 13 Exercise 13.4. **There are 8 solved examples given before this exercise. You must solve these examples before moving to the NCERT exercise questions. It will help you understand the concept easily. There are 17 type questions including 2 multiple-choice types questions given in **exercise 13.4 class 12 maths**.

**Benefits of NCERT Solutions for Class 12 Maths Chapter 13 Exercise 13.4:-**

There are 8 examples given before exercise 13.4 which you should solve first then try to solve exercise questions.

You are advised to solve more problems which will help you to get conceptual clarity.

**Class 12 maths chapter 13 exercise 13.4 solutions**are helpful in the revision of the important concepts**NCERT Solutions for Class 12 Maths Chapter 13 Exercise 13.4**can be used for reference.- Note- Don't buy any reference book as most of the questions in the board exams are asked directly from the NCERT textbook. Be thorough with the NCERT textbook.

NCERT solutions for class 12 maths chapter 13

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**Happy learning!!!**