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

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 = $frac { 6 }{ 30 }$ = $frac { 1 }{ 5 }$
Probability of getting a good bulb = $1-frac { 1 }{ 5 }$ = $frac { 4 }{ 5 }$
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 = $frac { 1 }{ 6 }$
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) $frac { 8 }{ 3 }$
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) $frac { 37 }{ 221 }$
(b) $frac { 5 }{ 13 }$
(c) $frac { 1 }{ 13 }$
(d) $frac { 2 }{ 13 }$
Solution:
n(S) = 52, n(A) = 4

Now E(X) = $frac { 2 }{ 13 }$
Option (d) is correct

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