NCERT Solutions for Class 11 Maths Chapter 14 Probability Ex 14.2
Ex 14.2 Class 11 Maths Question 1.
Which of the following cannot be valid assignment of probabilities for outcomes of sample space
S = {⍵1, ⍵2, ⍵3, ⍵4, ⍵5, ⍵6, ⍵7}
Solution:
(a) Sum of all the probabilities = 0.1 + 0.01 + 0.05 + 0.03 + 0.01 + 0.2 + 0.6 = 1.00
∴ Assignment of probabilities is valid.
(b) Sum of all the probabilities =1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7 = 7/7 = 1
∴ Assignment of probabilities is valid.
(c) Sum of all the probabilities = 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 = 2.8
Sum of all the probabilities is greater than 1.
∴ The assignment of probabilities is not valid.
(d) Probability of any event cannot be negative. Therefore, this assignment of probabilities is not valid.
(e) The last probability 15/14 is greater than 1. The probability of an event cannot be greater than 1.
∴ This assignment of probabilities is not valid.
Ex 14.2 Class 11 Maths Question 2.
A coin is tossed twice, what is the probability that at least one tail occurs?
Solution:
An experiment involves tossing a coin twice.
The sample space of the given experiment is given by:
S = {HH, HT, TH, TT}
Let E be the event of getting at least one tail.
Then, E = {HT, TH, TT}
∴ P(E) = n(E)/n(S) = 3/4
Ex 14.2 Class 11 Maths Question 3.
A die is thrown, find the probability of following events:
(i) A prime number will appear,
(ii) A number greater than or equal to 3 will appear,
(iii) A number less than or equal to one will appear,
(iv) A number more than 6 will appear,
(v) A number less than 6 will appear.
Solution:
An experiment involves throwing a die.
∴ The sample space of the experiment is given by S = {1, 2, 3, 4, 5, 6}
(i) Let E be the event that a prime number will appear. Then, E = {2, 3, 5}
∴ P(E) = n(E)/n(S) = 3/6 = 1/2
(ii) Let F be the event that a number ≥ 3 will appear.
Then, F = {3, 4, 5, 6}
∴ P(F) = n(F)/n(S) = 4/6 = 2/3
(iii) Let G be the event that a number ≤ 1 will appear.
Then, G = {1}.
∴ P(G) = n(G)/n(S) = 1/6
(iv) Let H be the event that a number more than 6 will appear.
Then, H = ⏀
∴ P(H) = n(H)/n(S) = 0/6 = 0
(v) Let I be the event that a number less than 6 will appear.
Then, I = {1, 2, 3, 4, 5}
∴ P(I) = n(I)/n(S) = 5/6
Ex 14.2 Class 11 Maths Question 4.
A card is selected from a pack of 52 cards.
(a) How many points are there in the sample space?
(b) Calculate the probability that the card is an ace of spades.
(c) Calculate the probability that the card is
(i) an ace
(ii) black card.
Solution:
(a) There are 52 cards in a pack.
⇒ Number of points in the sample space S = n(S) = 52
(b) Let E be the event of drawing an ace of spades.
There is only one ace of spade n(E) = 1 and n(S) = 52
∴ P(E) = n(E)/n(S) = 1/52
(c)
(i) Let F be the event of drawing an ace. There are 4 aces in a pack of 52 cards.
n(F) = 4, n(S) = 52
∴ P(F) = n(F)/n(S) = 4/52 = 1/13
(ii) Let G be the event of drawing a black card. There are 26 black cards.
n(G) = 26, n(S) = 52
∴ P(G) = n(G)/n(S) = 26/52 = 1/2
Ex 14.2 Class 11 Maths Question 5.
A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find the probability that the sum of numbers that turn up is
(i) 3
(ii) 12.
Solution:
An experiment involves tossing a coin marked 1 and 6 on either faces and rolling a die.
∴ The sample space of the experiment is given by:
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
(i) Let E be the event that the sum of numbers is 3.
Then, E = {(1, 2)} ⇒ n(E) = 1
n(S)= 12
∴ P(E) = n(E)/n(S) = 1/12
(ii) Let F be the event that the sum of numbers is 12.
Then, F = {(6, 6)} ⇒ n(F) = 1 and n(S) = 12
⇒ P(E) = n(F)/n(S) = 1/12
Ex 14.2 Class 11 Maths Question 6.
There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?
Solution:
There are 4 men and 6 women in the city council.
An experiment involves selecting a council member at random.
∴ n(S) = 10
Let E be the event that the selected council member is a woman.
Then, n(E) = 6
∴ P(E) = n(E)/n(S) = 6/10 = 3/5
Ex 14.2 Class 11 Maths Question 7.
A fair coin is tossed four times, and a person win Re 1 for each head and lose Rs 1.50 for each tail that turns up.
From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.
Solution:
An experiment involves tossing a fair coin four times. Therefore, the sample space of the given experiment is:
S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, HTHT, THTH, TTHH, THHT, HTTT, THTT, TTHT, TTTH, TTTT}
∴ n(S) = 16
According to the question, we have
Ex 14.2 Class 11 Maths Question 8.
Three coins are tossed once. Find the probability of getting
(i) 3 heads
(ii) 2 heads
(iii) at least 2 heads
(iv) at most 2 heads
(v) no head
(vi) 3 tails
(vii) exactly two tails
(viii) no tail
(ix) at most two tails
Solution:
An experiment involves tossing 3 coins.
∴ The sample space of the given experiment is:
S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
∴ n(S) = 8
Ex 14.2 Class 11 Maths Question 9.
If 2/11 is the probability of an event, what is the probability of the event ’not A’.
Solution:
Let P(A) = 2/11
P(not A) = 1 – P(A) = 1 – 2/11 = 9/11
Ex 14.2 Class 11 Maths Question 10.
A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is
(i) a vowel
(ii) a consonant.
Solution:
An experiment involves a letter chosen at random from the word ‘ASSASSINATION’ which consists 13 letters (6 vowels and 7 consonants).
∴ n(S) = 13.
(i) Let E be the event that chosen letter is a vowel.
Then, E = {A, A, A, I, I, O}
∴ n(E) = 6
⇒ P(E) = n(E)/n(S) = 6/13
(ii) Let F be the event that chosen letter is a consonant.
Then, F = {S, S, S, S, N, N, T}
⇒ P(F) = n(F)/n(S) = 7/13
Ex 14.2 Class 11 Maths Question 11.
In a lottery, a person choses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? [Hint: Order of the numbers is not important.]
Solution:
An experiment involves a lottery. A person chose six different natural numbers at random from 1 to 20.
∴ Sample points
= 20C6 = (20 × 19 × 18 × 17 × 16 × 15)/(1 × 2 × 3 × 4 × 5 × 6) = 38760
Let E be the event that chosen six numbers match with the six numbers already fixed by the lottery committee, i.e., winning the prize, in the game.
Then, n(E) = 6C6 = 1
∴ P(E) = n(E)/n(S) = 1/38760
Ex 14.2 Class 11 Maths Question 12.
Check whether the following probabilities P(A) and P(B) are consistently defined.
(i) P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6
(ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8
Solution:
(i) P(A ∩ B) must be less than or equal to P(A) and P(B).
∴ P(A ∩ B) = 0.6 > 0.5
∴ P(A) and P(B) are not defined consistently.
(ii) P(A ∩ B) = P(A) + P(B) – P(A ∪ B)
= 0.5 + 0.4 – 0.8
= 0.9 – 0.8 = 0.1
∴ P(A ∩ B) = 0.1 < 0.5 and P(A ∩ B) = 0.1 < 0.4
Thus, P(A) and P(B) are consistently defined.
Ex 14.2 Class 11 Maths Question 13.
Fill in the blanks in following table:
Solution:
(i) P(A∪ B) = P(A) + P(B) – P(A ∩ B)
= 1/3 + 1/5 – 1/15 = (5 + 3 – 1)/15 = 7/15
(ii) P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ 0.6 = 0.35 + P(B) – 0.25
∴ P(B) = 0.6 – 0.35 + 0.25 = 0.5
(iii) P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ 0.7 = 0.5 + 0.35 – P(A ∩ B)
∴ P(A ∩ B) = 0.5 + 0.35 – 0.7 = 0.15
Solution:
(i) P(A∪ B) = P(A) + P(B) – P(A ∩ B)
= 1/3 + 1/5 – 1/15 = (5 + 3 – 1)/15 = 7/15
(ii) P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ 0.6 = 0.35 + P(B) – 0.25
∴ P(B) = 0.6 – 0.35 + 0.25 = 0.5
(iii) P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ 0.7 = 0.5 + 0.35 – P(A ∩ B)
∴ P(A ∩ B) = 0.5 + 0.35 – 0.7 = 0.15
Ex 14.2 Class 11 Maths Question 14.
Given P(A) = 3/5 and P(B) = 1/5. Find P(A or B), if A and B are mutually exclusive events.
Solution:
If A and B are mutually exclusive events, then A ∩ B = ⏀
⇒ P(A ∩ B) = 0
∴ P(A ∪ B) = P(A) + P(B) = 3/5 + 1/5 = 4/5
Solution:
If A and B are mutually exclusive events, then A ∩ B = ⏀
⇒ P(A ∩ B) = 0
∴ P(A ∪ B) = P(A) + P(B) = 3/5 + 1/5 = 4/5
Ex 14.2 Class 11 Maths Question 15.
If E and F are events such that P(E) = 1/4, P(F) = 1/2 and P(E and F) = 1/8, find
(i) P(E or F),
(ii) P(not E and not F).
Solution:
(i) P(E or F) = P(E ∪ F)
= P(E) + P(F) – P(E ∩ F)
= ¼ + ½ − 1/8 = (2 + 4 – 1)/8 = 5/8
(ii) not E and not F = E’ ∩ F’ = (E ∪ F)’ (Using De Morgan’s Law)
∴ P(not E and not F) = P(E ∪ F)’
= 1 – P(E ∪ F) = 1 – 5/8 = 3/8
(i) P(E or F),
(ii) P(not E and not F).
Solution:
(i) P(E or F) = P(E ∪ F)
= P(E) + P(F) – P(E ∩ F)
= ¼ + ½ − 1/8 = (2 + 4 – 1)/8 = 5/8
(ii) not E and not F = E’ ∩ F’ = (E ∪ F)’ (Using De Morgan’s Law)
∴ P(not E and not F) = P(E ∪ F)’
= 1 – P(E ∪ F) = 1 – 5/8 = 3/8
Ex 14.2 Class 11 Maths Question 16.
Events E and F are such that P(not E or not F) = 0.25. State whether E and F are mutually exclusive.
Solution:
not E or not F = E’ ∪ F’ = (E ∩ F)’ (Using De Morgan’s Law)
∴ P(not E or not F) = P(E ∩ F)’ = 1 – P(E ∩ F)
⇒ 0.25 = 1 – P(E ∩ F)
⇒ P(E ∩ F) = 1 – 0.25 = 0.75 ≠ 0
∴ Events E and F are not mutually exclusive.
Solution:
not E or not F = E’ ∪ F’ = (E ∩ F)’ (Using De Morgan’s Law)
∴ P(not E or not F) = P(E ∩ F)’ = 1 – P(E ∩ F)
⇒ 0.25 = 1 – P(E ∩ F)
⇒ P(E ∩ F) = 1 – 0.25 = 0.75 ≠ 0
∴ Events E and F are not mutually exclusive.
Ex 14.2 Class 11 Maths Question 17.
A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine
(i) P(not A),
(ii) P(not B) and
(iii) P(A or B)
Solution:
(i) P(not A) = P(A’) = 1 – P(A) = 1 – 0.42 = 0.58
(ii) P(not B) = P(B’) = 1 – P(B) = 1 – 0.48 = 0.52
(iii) P(A or B) = P(A ∪ B)
= P(A) + P(B) – P(A ∩ B) = 0.42 + 0.48 – 0.16 = 0.74
(i) P(not A),
(ii) P(not B) and
(iii) P(A or B)
Solution:
(i) P(not A) = P(A’) = 1 – P(A) = 1 – 0.42 = 0.58
(ii) P(not B) = P(B’) = 1 – P(B) = 1 – 0.48 = 0.52
(iii) P(A or B) = P(A ∪ B)
= P(A) + P(B) – P(A ∩ B) = 0.42 + 0.48 – 0.16 = 0.74
Ex 14.2 Class 11 Maths Question 18.
In Class XI of a school, 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology.
Solution:
Let E and F be the events that students study Mathematics and Biology, respectively. Probability that students study Mathematics,
P(E) = 40/100 = 0.4
Probability that students study Biology,
P(F) = 30/100 = 0.3
Probability that students study both Mathematics and Biology,
P(E ∩ F) = 10/100 = 0.1
We have to find the probability that a student studies Mathematics or Biology, i.e., P(E ∪ F).
Now, P(E ∪ F) = P(E) + P(F) – P(E ∩ F) = 0.4 + 0.3 – 0.1 = 0.6
Solution:
Let E and F be the events that students study Mathematics and Biology, respectively. Probability that students study Mathematics,
P(E) = 40/100 = 0.4
Probability that students study Biology,
P(F) = 30/100 = 0.3
Probability that students study both Mathematics and Biology,
P(E ∩ F) = 10/100 = 0.1
We have to find the probability that a student studies Mathematics or Biology, i.e., P(E ∪ F).
Now, P(E ∪ F) = P(E) + P(F) – P(E ∩ F) = 0.4 + 0.3 – 0.1 = 0.6
Ex 14.2 Class 11 Maths Question 19.
In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both?
Solution:
Let E be the event that the student passes the first examination and F be the event that the student passes the second examination. Then, P(E) = 0.8, P(F) = 0.7 and P(E ∪ F) = 0.95
We know that
P(E ∪ F) = P(E) + P(F) – P(E ∩ F)
⇒ 0.95 = 0.8 + 0.7 – P(E ∩ F)
⇒ 0.95 = 1.5 – P(E ∩ F)
∴ P(E ∩ F) = 1.5 – 0.95 = 0.55
Solution:
Let E be the event that the student passes the first examination and F be the event that the student passes the second examination. Then, P(E) = 0.8, P(F) = 0.7 and P(E ∪ F) = 0.95
We know that
P(E ∪ F) = P(E) + P(F) – P(E ∩ F)
⇒ 0.95 = 0.8 + 0.7 – P(E ∩ F)
⇒ 0.95 = 1.5 – P(E ∩ F)
∴ P(E ∩ F) = 1.5 – 0.95 = 0.55
Ex 14.2 Class 11 Maths Question 20.
The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75, what is the probability of passing the Hindi examination?
Solution:
Let E be the event that student passes English examination and F be the event that the student passes Hindi examination.
Solution:
Let E be the event that student passes English examination and F be the event that the student passes Hindi examination.
Ex 14.2 Class 11 Maths Question 21.
In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that
(i) The student opted for NCC or NSS.
(ii) The student has opted neither NCC nor NSS.
(iii) The student has opted NSS but not NCC.
Solution:
Here, the total number of students, n(S) = 60.
Let E be the event that the student opted for NCC and F be the event that the student opted for NSS.
Then n(E) = 30, n(F) = 32 and n(E ∩ F) = 24
Related Links:
NCERT Solutions for Maths Class 9
NCERT Solutions for Maths Class 10
(i) The student opted for NCC or NSS.
(ii) The student has opted neither NCC nor NSS.
(iii) The student has opted NSS but not NCC.
Solution:
Here, the total number of students, n(S) = 60.
Let E be the event that the student opted for NCC and F be the event that the student opted for NSS.
Then n(E) = 30, n(F) = 32 and n(E ∩ F) = 24
Related Links:
NCERT Solutions for Maths Class 9
NCERT Solutions for Maths Class 10