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Textbook | NCERT |

Board | CBSE |

Category | NCERT Solutions |

Class | Class 12 |

Subject | Maths |

Chapter | Chapter 13 |

Exercise | Class 12 Maths Chapter 13 Probability Exercise 13.4 |

Number of Questions Solved | 18 |

Contents

## NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.4

**NCERT TEXTBOOK EXERCISES**

**Ex 13.4 Class 12 Maths Question 1.State which of the following are not the probability distributions of a random variable. Give reasons for your answer.**

**Solution:**

P (0) + P (1) + P (2) = 0.4 + 0.4 + 0.2 = 1

It is a probability distribution.

(ii) P (3) = -0.1 which is not possible.

Thus it is not a probability distribution.

(iii) P(-1)+P(0)+P(1) = 0.6 + 0.1 + 0.2 = 0.9≠1

Thus it is not a probability distribution.

(iv) P (3) + P (2) + P (1) + P (0) + P (-1)

= 0.3 + 0.2 + 0.4 + 0.1 + 0.05 = 1.05≠1

Hence it is not a probability distribution.

**Ex 13.4 Class 12 Maths Question 2.An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X ?. Is X a random variable?**

Solution:

These two balls may be selected as RR, RB, BR, BB, where R represents red and B represents black ball, variable X has the value 0,1,2, i.e., there may be no black balls, may be one black ball, or both the balls are.black. Yes , X is a random variable.

**Ex 13.4 Class 12 Maths Question 3.Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?**

Solution:

For one coin, S = {H,T}

n (S) = 2, Let A represent Head

∴ A = {H},n(A) = 1

**Ex 13.4 Class 12 Maths Question 4.Find the probability distribution of**

(a) number of heads in two tosses of a coin.

(b) number of tails in the simultaneous tosses of three coins.

(c) number of heads in four tosses of a coin.

Solution:

(a) When two tosses of a coin are there sample space

= {TT, TH, HT, HH}

Zero success => No heads => Two tails (TT)

**Ex 13.4 Class 12 Maths Question 5.Find the probability distribution of the number of successes in two tosses of a die where a success Is defined as**

(i) number greater than 4

(ii) six appears on at least one die

**Solution:**

S = (1, 2, 3,4, 5,6},n(S) = 6

(i) Let A be the set of favorable events.

A = {5,6),n(A) = 2

**Ex 13.4 Class 12 Maths Question 6.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.**

Solution:

There are 30 bulbs which include 6 defective bulbs

Probability of getting a defective bulb = 630 = 15

Probability of getting a good bulb = 1−15 = 45

Let X denotes variable of defective bulbs in a sample of 4 bulb

**Ex 13.4 Class 12 Maths Question 7.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 tads.**

**Solution:**

Let p represents the appearance of tail.

q represents the appearance of head.

Now q = 3p As p + q = 1 => p + 3p = 1

**Ex 13.4 Class 12 Maths Question 8A random variable X has the following probability distribution:**

Determine

(i) k

(ii) P(X<3) (iii) P(X>6)

(iv) P(0<X<3)

Solution:

(i) Sum of probabilities = 1

**Ex 13.4 Class 12 Maths Question 9.The random variable X has a probability distribution P (X) of the following form, where k is some number**

(a) Determine the value of k

(b) FindP(X<2),P (X≤2), P(X≥2)

**Solution:**

(a) Sum of probabilities = 1

k + 2k + 3k = 1 or 6k = 1,k = 16

The probability distribution is as given below

**Ex 13.4 Class 12 Maths Question 10.Find the mean number of heads in three tosses of a fair coin.**

**Solution:**

S = {H,T},n(S) = 2

Let A denotes the appearance of head on a toss A = {H}

**Ex 13.4 Class 12 Maths Question 11.Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.**

**Solution:**

Two dice thrown simultaneously is the same the die thrown 2 times.

Let S= {1,2,3,4,5,6},n(S) = 6

Let A denotes the number 6

**Ex 13.4 Class 12 Maths Question 12.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)**

**Solution:**

There are six numbers 1,2,3,4,5,6 one of them is selected in 6 ways

When one of the numbers has been selected, 5 numbers are left, one number out of 5 may be select in 5 ways

∴ No. of ways of selecting two numbers without replacement out of 6 positive integers = 6 x 5 = 30

**Ex 13.4 Class 12 Maths Question 13.Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.**

**Solution:**

n (S) = 36

Let A denotes the sum of the numbers = 2

B denotes the sum of the numbers = 3

C denotes the sum of the numbers = 4

D denotes the sum of the numbers = 5

E denotes the sum of the numbers = 6

F denotes the sum of the numbers = 7

G denotes the sum of the numbers = 8

H denotes the sum of the numbers = 9

I denotes the sum of the numbers = 10

J denotes the sum of the numbers = 11

K denotes the sum of the numbers = 12

**Ex 13.4 Class 12 Maths Question 14.A class has 15 students whose ages are 14,17, 15,14,21,17,19,20,16,18,20,17,16,19 and 20 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 X ? Find mean, variance and standard deviation of X?**

**Solution:**

There are 15 students in a class. Each has the same chance of being choosen.

The probability of each student to be selected

**Ex 13.4 Class 12 Maths Question 15.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).**

Solution:

Here the variable values are 1 and 0 and the probability of occurrence is 70% = 0.7 and 30% = 0.3

Probability distribution is

Choose the correct answer in each of the following:

**Ex 13.4 Class 12 Maths Question 16.The mean of the number obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is**

(a) 1

(b) 2

(c) 5

(d) 83

**Solution:**

Mean 2

Option (b) is correct

**Ex 13.4 Class 12 Maths Question 17.Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. What is the value of E (X)?**

(a) 37221

(b) 513

(c) 113

(d) 213

**Solution:**

n(S) = 52, n(A) = 4

Now E(X) = 213

Option (d) is correct

## NCERT Solutions for Class 12 Maths Chapter 13 Probability Exercise 13.4 PDF

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