These NCERT Solutions for Class 12 Maths Chapter 13 Probability Miscellaneous Exercise Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-13-miscellaneous-exercise/
NCERT Solutions for Class 12 Maths Chapter 13 Probability Miscellaneous Exercise
Question 1.
A and B are two events such that P (A) ≠ 0. Find P(B|A), if
i. A is a subset of B
ii. A ∩ B = Φ
Solution:
Question 2.
A couple has two children.
Find the probability that both children are males, if it is known that atleast one of the children is male.
Solution:
The sample space, S = {MM, MF, FM, FF},
where M denote male and F denote female.
Let A: both children are males
B : atleast one child is a male
A = {MM},
B = {MM, MF, FM}
\(\mathrm{A} \cap \mathrm{B}=\{\mathrm{MM}\}, \mathrm{P}(\mathrm{A})=\frac{1}{4}\)
\(P(B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}\)
Question 3.
If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?
Solution:
A leap year contains 366 days = 52 weeks + 2 days
The last 2 days can be
i. Monday, Tuesday
ii. Tuesday, Wednesday
iii. Wednesday, Thursday
iv. Thursday, Friday
v. Friday, Saturday
vi. Saturday, Sunday
vii. Sunday, Monday
Of these seven possibilities, (i) & (ii) are favourable to 53 Tuesdays.
∴ P(53 Tuesday) = \(\frac { 2 }{ 7 }\)
Question 4.
An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes.
Solution:
Let p be the probability of a success and q the probability of failure.
Then p + q – 1 and p = 2q
Solving p = \(\frac { 2 }{ 3 }\) and q = \(\frac { 1 }{ 3 }\)
Let X be the number of success.
Then X is a binomial distribution with
Question 5.
How many times must a man toss a fair coin so that the probability of having atleast one head is more than 90%?
Solution:
Tossing a coin many times is a Bernoulli trial. Here success is obtaining a Head.
∴ P = \(\frac { 1 }{ 2 }\)
q = 1 – p = 1 – \(\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 2 }\)
Let X be the number of heads obtained
Then X is a binomial distribution B(n, \(\frac { 1 }{ 2 }\))
We know 21 = 2, 2² = 4, 2³ = 8, 24 = 16, 25 = 32 and so on
Hence the minimum value of n is 4 i.e. n > 4
i.e. the man has to toss the coin atleast 4 times.
Question 6.
In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six, Find the expected value of the amount he wins/loses.
Solution:
The game ends in the following ways.
i. The man gets 6 in 1st throw. In this case, he earns ₹ 1.
P(getting 6 ¡n 1st throw) = \(\frac { 1 }{ 6 }\)
ii. The man does not get 6 in 1nd throw and 6 in 2nd throw. In this case he earns ₹ 0
(In 1st throw, he earns ₹ 1 and in 2nd throw he loses ₹ 1)
P(not getting 6 on 1st throw & 6 in 2nd throw) = (\(\frac { 5 }{ 6 }\))(\(\frac { 1 }{ 6 }\)) = \(\frac { 5 }{ 6 }\)
Question 7.
Suppose we have four boxes A, B, C and D containing coloured marbles as given below:
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A? box B?
Solution:
Let E1 : selecting box A
E2 : selecting box B
E3 : selecting box C
E4 : selecting box D
A : selecting a red ball
E1, E2, E3 and E4 are mutually exclusive and exhaustive events.
∴ \(\mathrm{P}\left(\mathrm{E}_{1}\right)=\mathrm{P}\left(\mathrm{E}_{2}\right)=\mathrm{P}\left(\mathrm{E}_{3}\right) \dot{\mathrm{P}}\left(\mathrm{E}_{4}\right)=\frac{1}{4}\)
\(\mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{1}\right)=\frac{1}{10}, \quad \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{2}\right)=\frac{6}{10}\)
Question 8.
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
Solution:
E1 : a red ball is transferred from bag I to bag II
E2 : a black ball is transferred from bag I to bag II
A : a red ball is taken from bag II after transferring a ball
E1 and E2 are mutually exclusive and exhaustive events
Question 9.
If A B are two events such that P(A) ≠ 0 and P(B|A) = 1, then
a. A ∩ B
b. B ∩ A
c. B = Φ
d. A = Φ
Solution:
Question 10.
If P(A|B) > P(A), then which of the following is correct:
a. P(B | A) < P(B)
b. P(A ∩ B) < P(A). P(B) c. P(B | A) > P(B)
d. P(B | A) = P(B)
Solution:
Question 11.
If A and B are any two events such that P(A) + P(B) – P(A and B) = P(A), then
a. P(B|A) = 1
b. P(A|B) = 1
c. P(B|A) = 0
d. P(A|B) = 0
Solution: