Ex 13.5 Class 12 Maths Question 1.
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?
Solution:
There are 3 odd numbers on a die
∴ Probability of getting an odd number on a die = $\frac { 3 }{ 6 }$

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NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q1.1

Ex 13.5 Class 12 Maths Question 2.
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item ?
Solution:
Probability of getting one defective item = 5%
= $\frac { 5 }{ 100 }$
= $\frac { 1 }{ 20 }$
Probability of getting a good item = $1-\frac { 1 }{ 20 }$ = $\frac { 19 }{ 20 }$
A sample of 10 item include not more than one defective item.
=> sample contains at most (me defective item Its probability = P (0) + P (1)
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q2.1

Ex 13.5 Class 12 Maths Question 3.
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability, of two successes.
Solution:
n(S) = 36, A = {11,22,33,44,55,66}
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q3.1

Ex 13.5 Class 12 Maths Question 4.
Five cards are drawn successively with replacement from a well- shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is spade?
Solution:
S = {52 cards}, n (S) = 52
Let A denotes the favourable events
A= {13 spade}, n(A)= 13
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q4.1

Ex 13.5 Class 12 Maths Question 5.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use
Solution:
Probability that a bulb gets fuse after 150 days of its use = 0.05
Probability that the bulb will not fuse after 150 days of its use = 1 – 0.05 = 0.95
(i) Probability that no bulb will fuse after 150
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q5.1

Ex 13.5 Class 12 Maths Question 6.
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four bails are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Solution:
S = {0,1,2,3,4,5,6,7,8,9},n(S) = 10
Let A represents that the ball is marked with the digit 0.
A = {0}, n(A) = 1
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q6.1

Ex 13.5 Class 12 Maths Question 7.
In an examination, 20 questions of true – false type are asked. Suppose a student tosses fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true,’ if it falls tails, he answers “ false’. Find the probability that he answers at least 12 questions correctly.
Solution:
Probability that student answers a question true = $\frac { 1 }{ 2 }$
i.e., when a coin is thrown, probability that a head is obtained = $\frac { 1 }{ 2 }$
Probability that his answer is false = $1-\frac { 1 }{ 2 }$ = $\frac { 1 }{ 2 }$
Probability that his answer at least 12 questions correctly = P (12) + P (13) + P (14) +…….. P (20)
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q7.1

Ex 13.5 Class 12 Maths Question 8
Suppose X has a binomial distribution $B\left( 6,\frac { 1 }{ 2 } \right)$ . Show that X = 3 is the most likely outcome.
(Hint: P (X = 3) is the maximum among all P (Xi), xi. = 0,1,2,3,4,5,6)
Solution:
${ \left( \frac { 1 }{ 2 } +\frac { 1 }{ 2 } \right) }^{ 6 }$
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q8.1

Ex 13.5 Class 12 Maths Question 9.
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Solution:
P = $\frac { 1 }{ 3 }$ . q = 1 – P = $1-\frac { 1 }{ 3 }$ = $\frac { 2 }{ 3 }$
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q9.1

Ex 13.5 Class 12 Maths Question 10.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac { 1 }{ 100 }$ . What is the probability that he will win a prize?
(a) at least once,
(b) exactly once,
(c) at least twice?
Solution:
Probability that the person wins the prize = $\frac { 1 }{ 100 }$
Probability of losing = $1-\frac { 1 }{ 100 }$ = $\frac { 99 }{ 100 }$
(a) Probability that he loses in all the loteries
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q10.1

Ex 13.5 Class 12 Maths Question 11.
Find the probability of getting 5 exactly twice in 7 throws of a die.
Solution:
S = {1,2,3,4,5,6},n(S) = 6
A = {5} => n(A) = 1
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q11.1

Ex 13.5 Class 12 Maths Question 12.
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Solution:
When a die is thrown,
Probabiltiy of getting a six = $\frac { 1 }{ 6 }$
Probabiltiy of not getting a six = $1-\frac { 1 }{ 6 }$ = $\frac { 5 }{ 6 }$
Probabiltiy of getting at most 2 sixes in 6 throws of a single die = P (0) + P (1) + P (2)
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q12.1

Ex 13.5 Class 12 Maths Question 13.
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles 9 are defective?
Solution:
p = $\frac { 1 }{ 100 }$ = $\frac { 1 }{ 10 }$
q = 1 – p = $1-\frac { 1 }{ 10 }$ = $\frac { 9 }{ 10 }$
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q13.1

Ex 13.5 Class 12 Maths Question 14.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(a) ${ 10 }^{ -1 }$
(b) ${ \left( \frac { 1 }{ 2 } \right) }^{ 5 }$
(c) ${ \left( \frac { 9 }{ 10 } \right) }^{ 5 }$
(d) $\frac { 9 }{ 10 }$
Solution:
p = $\frac { 1 }{ 10 }$
q = $\frac { 9 }{ 10 }$ n = 5, r = 0, P(X=0) = ${ \left( \frac { 9 }{ 10 } \right) }^{ 5 }$
Option (c) is correct

Ex 13.5 Class 12 Maths Question 15.
The probability that a student is not a swimmer is $\frac { 1 }{ 5 }$ . Then the probability that out of five students, four are swimmers is:
vedantu class 12 maths Chapter 13 Probability 15
Solution:
p = $\frac { 4 }{ 5 }$ , q = $\frac { 1 }{ 5 }$ , n = 5,r = 4
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q15.1

Option (a) is true

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