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

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## 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 8
A 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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