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NCERT Solutions for Class 11 Maths Chapter 16 Probability are provided here. Provides NCERT solutions in simple, comprehensive and step by step solutions for all questions. these NCERT solutions are prepared by an expert team of Careers360 keeping in mind the latest syllabus of CBSE 2023.Students can practice these NCERT solutions to command all the concepts which overall lead to scoring well in the board as well as in competitive exams.
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There are 44 questions in 3 exercises in the NCERT textbook. First, try to solve it on your own, if you are not able to do so, you can take help of class 11 maths chapter 16 question answer. This chapter is very important for CBSE class 11 final examination as well as in the various competitive exams like JEE Main, BITSAT etc. class 11 maths chapter 16 NCERT solutions is useful to study advanced topics like probability distribution, stochastic process, mathematics statics and probability(MSP). Here you will get NCERT solutions for class 11 also.
Also Read :
Probability:
Probability is calculated as the number of favourable outcomes in an event divided by the total number of possible outcomes.
Probability = Number of Favourable Outcomes / Total Number of Outcomes
Event:
An event is a subset of the sample space S. An empty set is known as the Impossible event.
For any random experiment with sample space S, the probability P is a real-valued function where:
P(E) ≥ 0 for any event E
P(S) = 1
Mutually Exclusive Events:
If events E and F are mutually exclusive, then the probability of their union is the sum of their individual probabilities.
P(E ∪ F) = P(E) + P(F)
Equally Likely Outcomes:
Equally likely outcomes are outcomes with the same probability of occurring.
Let S be a finite sample space with equally likely outcomes, and A be an event. The probability of event A is calculated as:
P(A) = Number of Favourable Outcomes (n(A)) / Total Number of Outcomes (n(S))
Formulas for Combining Events:
Event P or Q corresponds to the set union P ∪ Q.
Event P and Q corresponds to the set intersection P ∩ Q.
Event P and not Q corresponds to the set difference P - Q.
Mutually Exclusive Events:
Events P and Q are mutually exclusive if their intersection is the empty set:
P ∩ Q = φ (empty set)
Exhaustive and Mutually Exclusive Events:
Events P1, P2, ..., Pn are exhaustive and mutually exclusive if:
Their union covers the entire sample space: P1 ∪ P2 ∪ ... ∪ Pn = S
They have no common outcomes: Ei ∩ Ej = φ for all i ≠ j
Free download NCERT Solutions for Class 11 Maths Chapter 16 Probability for CBSE Exam.
Probability class 11 ncert solutions - Exercise: 16.1
Question:1 Describe the sample space for the indicated experiment.
A coin is tossed three times.
Answer:
Let H denote Heads and T denote Tails.
For each toss, there are two possible outcomes = H or T
The required sample space is:
S = {HHH, HHT, HTH, THH, TTH, HTT, THT, TTT}
Question:2 Describe the sample space for the indicated experiment.
Answer:
When a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}
The required sample space is:
S = { = 1,2,3,4,5,6}
or S = {(1,1), (1,2), (1,3), ..., (1,6), (2,1), (2,2), ..., (2,6), ..., (6, 1), (6, 2), ..., (6,6)}
Question:3 Describe the sample space for the indicated experiment.
Answer:
Let H denote Heads and T denote Tails.
For each toss, there are two possible outcomes = H or T
The required sample space is:
S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT}
Question:4 Describe the sample space for the indicated experiment.
A coin is tossed and a die is thrown.
Answer:
Let H denote Heads and T denote Tails.
For each toss, there are two possible outcomes = H or T
And,
When a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}
The required sample space is:
S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
Question:5 Describe the sample space for the indicated experiment.
A coin is tossed and then a die is rolled only in case a head is shown on the coin.
Answer:
Let H denote Heads and T denote Tails.
For each toss, there are two possible outcomes = H or T
For H, when a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}
The required sample space is:
S = {H1, H2, H3, H4, H5, H6, T}
Question:6 Describe the sample space for the indicated experiment.
Answer:
Let X denote the event Room X is selected, Y denote the event Room Y is selected.
B1, B2 denote the event a boy is selected and G1, G2 denote the event a girl is selected from room X.
B3 denotes the event a boy is selected and G3, G4, G5 denote the event a girl is selected from Room Y.
The required sample space is:
S = {XB1 , XB2 , XG1 , XG2 , YB3 , YG3 , YG4 , YG5 }
Question:7. Describe the sample space for the indicated experiment.
Answer:
Let, R denote the event the red die comes out,
W denote the event the white die comes out,
B denote the event the Blue die is chosen
When a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}
The required sample space is:
S = {R1, R2, R3, R4, R5, R6, W1, W2, W3, W4, W5, W6, B1, B2, B3, B4, B5, B6}
Question:8(i) An experiment consists of recording boy–girl composition of families with 2 children.
Answer:
Let B denote the event a boy is born,
G denote the event a girl is born
The required sample space with a boy or girl in the order of their births is:
S = {BB, BG, GB, GG}
Question: 8(ii). An experiment consists of recording boy–girl composition of families with 2 children.
What is the sample space if we are interested in the number of girls in the family?
Answer:
(ii) In a family with two child, there can be only three possible cases:
no girl child, 1 girl child or 2 girl child
The required sample is:
S = {0, 1, 2}
Question:9 A box contains red and identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.
Answer:
Given, Number of red balls =1
Number of white balls = 3
Let R denote the event that the red ball is drawn.
And W denotes the event that a white ball is drawn.
Since two balls are drawn at random in succession without replacement,
if the first ball is red, the second ball will be white. And if the first ball is white, second can be either of red and white
The required sample space is:
S = {RW, WR, WW}
Answer:
Let H denote the event that Head occurs and T denote the event that Tail occurs.
For T in first toss, the possible outcomes when a die is thrown = {1, 2, 3 ,4 ,5 ,6}
The required sample space is :
S = {HH, HT, T1, T2, T3, T4, T5, T6}
Answer:
Let D denote the event the bulb is defective and N denote the event the bulb is non-defective
The required sample space is:
S = {DDD, DDN, DND, NDD, DNN, NDN, NND, NNN}
Answer:
Possible outcomes when a coin is tossed = {H,T}
Possible outcomes when a die is thrown = {1,2,3,4,5,6}
When T occurs, experiment is finished. S1 = {T}
When H occurs, a die is thrown.
If the outcome is odd ({1,3,5}), S2 = {H1, H3, H5}
If the outcome is even({2,4,6}), the die is thrown again.,
S3 = {H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}
The required sample space is:
S = {T, H1, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}
Answer:
Given, two slips are drawn from the box, one after the other, without replacement.
Let 1, 2, 3, 4 denote the event that 1, 2, 3, 4 numbered slip is drawn respectively.
When two slips are drawn without replacement, the first event has 4 possible outcomes and the second event has 3 possible outcomes
S = {(1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2), (4,3)}
Answer:
Possible outcomes when a die is thrown = {1,2,3,4,5,6}
Possible outcomes when a coin is tossed = {H,T}
If the number on the die is even {2,4,6}, the coin is tossed once.
S1 = {2H, 2T, 4H, 4T, 6H, 6T}
If the number on the die is odd {1,3,5}, the coin is tossed twice.
S2 = {1HH, 1HT, 1TH, 1TT, 3HH, 3HT, 3TH, 3TT, 5HH, 5HT, 5TH, 5TT}
The required sample space is:
S = {1HH, 1HT, 1TH, 1TT, 2H, 2T, 3HH, 3HT, 3TH, 3TT, 4H, 4T, 5HH, 5HT, 5TH, 5TT, 6H, 6T}
Answer:
Possible outcomes when a coin is tossed = {H,T}
Possible outcomes when a die is thrown = {1,2,3,4,5,6}
Let R1 and R2 denote the event that a red ball is drawn
and B1, B2, B3 denote the event that a blue ball is drawn
If H occurs, a die is thrown.
S1 = {H1, H2, H3, H4, H5, H6}
If T occurs, a ball from a box which contains red and black balls is drawn.
S2 = {TR1 , TR2 , TB1 , TB2 , TB3}
The required sample space is:
S = {TR1 , TR2 , TB1 , TB2 , TB3 , H1, H2, H3, H4, H5, H6}
Question:16 A die is thrown repeatedly untill a six comes up. What is the sample space for this experiment?
Answer:
Given, a die is thrown repeatedly untill a six comes up.
Possible outcomes when a die is thrown = {1,2,3,4,5,6}
In the experiment 6 may come up on the first throw, or the 2nd throw, or the 3rd throw and so on till 6 is obtained.
The required sample space is:
S = {6, (1,6), (2,6), (3,6), (4,6), (5,6), (1,1,6), (1,2,6), ..., (1,5,6), (2,1,6). (2,2,6), ..., (2,5,6), ..., (5,1,6), (5,2,6), ... }
Class 11 maths chapter 16 question answer - Exercise 16.2
Answer:
When a die is rolled, the sample space of possible outcomes:
S = {1, 2, 3, 4, 5, 6}
Now,
E = event that the die shows 4 = {4}
F = event that the die shows even number = {2, 4, 6}
E F = {4} {2, 4, 6}
= {4}
Hence E and F are not mutually exclusive event.
Question:2(i) A die is thrown. Describe the following events:
Answer:
When a die is rolled, the sample space of possible outcomes:
S = {1, 2, 3, 4, 5, 6} or {x : x N, x<7}
Given, A : a number less than 7
As every number on a die is less than 7
A = {1, 2, 3, 4, 5, 6} = S
Question:2(ii) A die is thrown. Describe the following events:
Answer:
When a die is rolled, the sample space of possible outcomes:
S = {1, 2, 3, 4, 5, 6} or {x : x N, x<7}
Given, B: a number greater than 7
As no number on the die is greater than 7
B =
Question:2(iii) A die is thrown. Describe the following events:
Answer:
When a die is rolled, the sample space of possible outcomes:
S = {1, 2, 3, 4, 5, 6} or {x : x N, x<7}
Given, C : a multiple of 3
C = {3, 6}
Question:2(iv) A die is thrown. Describe the following events:
Answer:
When a die is rolled, the sample space of possible outcomes:
S = {1, 2, 3, 4, 5, 6} or {x : x N, x<7}
Given, D : a number less than 4
D = {1, 2, 3}
Question:2(v) A die is thrown. Describe the following events:
E: an even multiple greater than 4
Answer:
When a die is rolled, the sample space of possible outcomes:
S = {1, 2, 3, 4, 5, 6} or {x : x N, x<7}
Given, E : an even number greater than 4
S1 = Subset of S containing even numbers = {2,4,6}
Therefore , E = {6}
Question:2(vi). A die is thrown. Describe the following events:
Answer:
When a die is rolled, the sample space of possible outcomes:
S = {1, 2, 3, 4, 5, 6} or {x : x N, x<7}
Given, F : a number not less than 3
F = {x: x S, x 3 } = {3, 4, 5, 6}
Question:.(vi) A die is thrown. Describe the following events:
Answer:
A = {1, 2, 3, 4, 5, 6}
B=
A B = { 1, 2, 3, 4, 5, 6 } = {1, 2, 3, 4, 5, 6}
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
A = {1, 2, 3, 4, 5, 6}
B=
A B = {1, 2, 3, 4, 5, 6} =
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
B=
C= {3, 6}
B C = { 3, 6 } = {3, 6}
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
E = {6}
F = {3, 4, 5, 6}
E F = { 6 } {3, 4, 5, 6 } = {6}
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
D = {1, 2, 3}
E = {6}
D E = {1, 2, 3} {6} = (As nothing is common in these sets)
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
A = {1, 2, 3, 4, 5, 6}
C = {3, 6}
A - C = {1, 2, 3 , 4, 5, 6 } - { 3 , 6 } = {1, 2, 4, 5}
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
D = {1, 2, 3}
E = {6}
D - E = {1, 2, 3} - {6} = {1, 2, 3}
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
E = {6}
F = {3, 4, 5, 6}
F' = {3, 4, 5, 6}' = S - F = {1, 2}
E F' = {6} {1, 2} =
Question:2.(vi) A die is thrown. Describe the following events:
Answer:
F = {3, 4, 5, 6}
F' = {3, 4, 5, 6}' = S - F = {1, 2}
Question:3(a) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
Answer:
Sample space when a die is rolled:
S = {1, 2, 3, 4, 5, 6}
Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]
E = [ {(x,y): x,y S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}
Now,
A : the sum is greater than 8
Possible sum greater than 8 are 9, 10, 11 and 12
A = [ {(a,b): (a,b) E, a+b>8 } ] = {(3,6), (4,5), (5, 4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}
Question:3(b) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
Answer:
Sample space when a die is rolled:
S = {1, 2, 3, 4, 5, 6}
Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]
E = [ {(x,y): x,y S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}
Now,
B: 2 occurs on either die
Hence the number 2 can come on first die or second die or on both the die simultaneously.
B = [ {(a,b): (a,b) E, a or b = 2 } ] = {(1,2), (2,2), (3, 2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)}
Question:3(c). An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
the sum is at least and a multiple of
Answer:
Sample space when a die is rolled:
S = {1, 2, 3, 4, 5, 6}
Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]
E = [ {(x,y): x,y S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}
Now,
C: the sum is at least 7 and a multiple of 3
The sum can only be 9 or 12.
C = [ {(a,b): (a,b) E, a+b>6 & a+b = 3k, k I} ] = {(3,6), (6,3), (5, 4), (4,5), (6,6)}
Question:3(d). An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
Which pairs of these events are mutually exclusive?
Answer:
For two elements to be mutually exclusive, there should not be any common element amongst them.
Also, A = { (3,6) , (4,5), (5, 4), (6,3) , (4,6), (5,5), (6,4), (5,6), (6,5), (6,6) }
B = {(1,2), (2,2), (3, 2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)}
C = { (3,6), (6,3), (5, 4), (4,5), (6,6) }
Now, A B = (no common element in A and B)
Hence, A and B are mutually exclusive
Again, B C = (no common element in B and C)
Hence, B and C are mutually exclusive
Again, C A = {(3,6), (6,3), (5, 4), (4,5), (6,6)}
Therefore,
A and B, B and C are mutually exclusive.
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Now,
A = Event that three heads show up = {HHH}
B = Event that two heads and one tail show up = {HHT, HTH, THH}
C = Event that three tails show up = {TTT}
D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}
(i). For two elements X and Y to be mutually exclusive, X Y =
A B = {HHH} {HHT, HTH, THH} = ; Hence A and B are mutually exclusive.
B C = {HHT, HTH, THH} {TTT} = ; Hence B and C are mutually exclusive.
C D = {TTT} {HHH, HHT, HTH, HTT} = ; Hence C and D are mutually exclusive.
D A = {HHH, HHT, HTH, HTT} {HHH} = {HHH} ; Hence D and A are not mutually exclusive.
A C = {HHH} {TTT} = ; Hence A and C are mutually exclusive.
B D = {HHT, HTH, THH} {HHH, HHT, HTH, HTT} = {HHT, HTH} ; Hence B and D are not mutually exclusive.
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Now,
A = Event that three heads show up = {HHH}
B = Event that two heads and one tail show up = {HHT, HTH, THH}
C = Event that three tails show up = {TTT}
D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}
(ii).If an event X has only one sample point of a sample space, it is called a simple event.
A = {HHH} and C = {TTT}
Hence, A and C are simple events.
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, THH, TTH, TTT}
Now,
A = Event that three heads show up = {HHH}
B = Event that two heads and one tail show up = {HHT, HTH, THH}
C = Event that three tails show up = {TTT}
D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}
(iv). If an event has more than one sample point, it is called a Compound event.
B = {HHT, HTH, THH} and D = {HHH, HHT, HTH, HTT}
Hence, B and D are compound events.
Question:5(i) Three coins are tossed. Describe
Two events which are mutually exclusive.
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, THH, TTH, TTT}
(i)
A = Event that three heads show up = {HHH}
B = Event that three tails show up = {TTT}
A B = {HHH} {TTT} = ; Hence A and B are mutually exclusive.
Question:5(ii) Three coins are tossed. Describe
Three events which are mutually exclusive and exhaustive.
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting no tails = {HHH}
B = Getting exactly one tail = {HHT, HTH, THH}
C = Getting at least two tails = {HTT, THT, TTH}
Clearly, A B = ; B C = ; C A =
Since (A and B), (B and C) and (A and C) are mutually exclusive
Therefore A, B and C are mutually exclusive.
Also,
A B C = S
Hence A, B and C are exhaustive events.
Hence, A, B and C are three events which are mutually exclusive and exhaustive.
Question:5(iii). Three coins are tossed. Describe
Two events, which are not mutually exclusive.
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting at least one head = {HHH, HHT, HTH, THH, TTH}
B = Getting at most one head = {TTH, TTT}
Clearly, A B = {TTH}
Hence, A and B are two events which are not mutually exclusive.
Question:5.(iv) Three coins are tossed. Describe
Two events which are mutually exclusive but not exhaustive.
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting exactly one head = {HTT, THT, TTH}
B = Getting exactly one tail = {HHT, HTH, THH}
Clearly, A B =
Hence, A and B are mutually exclusive.
Also, A B S
Hence, A and B are not exhaustive.
Question:5.(v) Three coins are tossed. Describe
Three events which are mutually exclusive but not exhaustive
Answer:
Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting exactly one tail = {HHT, HTH, THH}
B = Getting exactly two tails = {HTT, TTH, THT}
C = Getting exactly three tails = {TTT}
Clearly, A B = ; B C = ; C A =
Since (A and B), (B and C) and (A and C) are mutually exclusive
Therefore A, B and C are mutually exclusive.
Also,
A B C = {HHT, HTH, THH, HTT, TTH, THT, TTT} S
Hence A, B and C are not exhaustive events.
Question:6.(i) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice .
Describe the events
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 6}
= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
(i) Therefore, A'= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
= B : getting an odd number on the first die.
Question:6.(ii) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice .
Describe the events
not B
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 6}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
(ii) Therefore, B'= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
= A : getting an even number on the first die.
Question:âââââââ6.(iii) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice .
Describe the events
A or B
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 6}
= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 6}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
(iii) A or B = A B = {(1,1), (1,2) .... (1,6), (3,1), (3,2).... (3,6), (5,1), (5,2)..... (5,6), (2,1), (2,2)..... (2,6), (4,1), (4,2)..... (4,6), (6,1), (6,2)..... (6,6)} = S
Question:6.(iv) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice
Describe the events
A and B
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 6}
= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 6}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
(iii) A and B = A B = A A' = (From (ii))
Question:6.(v) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice
Describe the events
A but not C
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 6}
= { (2, 1), (2,2), (2,3) , (2,4), (2,5), (2,6), (4,1) , (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
C: getting the sum of the numbers on the dice 5
The possible sum are 2,3,4,5
C = {(a,b): 2 a + b 5} = { (1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}
(v) A but not C = A - C = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
Question:.(vi) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice
Describe the events
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 6}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
C: getting the sum of the numbers on the dice 5
The possible sum are 2,3,4,5
C = {(a,b): 2 a + b 5} = { (1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1) }
(vi) B or C = B C = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3) , (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1) , (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
Question:6.(vii) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice
Describe the events
B and C
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 6}
= { (1,1), (1,2), (1,3), (1,4) , (1,5), (1,6), (3,1), (3,2) , (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
C: getting the sum of the numbers on the dice 5
The possible sum are 2,3,4,5
C = {(a,b): 2 a + b 5} = { (1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}
(vi) B and C = B C = {(1, 1), (1,2), (1,3), (1,4), (3,1), (3,2)}
Question:6.(viii) Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice
Describe the events
Answer:
Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 6}
= { (2, 1), (2,2), (2,3) , (2,4), (2,5), (2,6), (4,1) , (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 6}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
C: getting the sum of the numbers on the dice 5
The possible sum are 2,3,4,5
C = {(a,b): 2 a + b 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3) , (3,1), (3,2), (4,1) }
(viii) A B' C' = A A C' (from (ii))
= A C' = A - C = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
Question:7.(i) Refer to question 6 above, state true or false: (give reason for your answer)
A and B are mutually exclusive
Answer:
Here,
A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
B = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) }
(i) X and Y are mutually exclusive if and only if X Y =
A B = , since A and B have no common element amongst them.
Hence, A and B are mutually exclusive. TRUE
Question:7.(ii) Refer to question 6 above, state true or false: (give reason for your answer)
A and B are mutually exclusive and exhaustive
Answer:
Here,
A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
B = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) }
(ii) X and Y are mutually exclusive if and only if X Y =
A B = , since A and B have no common element amongst them.
Hence, A and B are mutually exclusive.
Also,
A B = {(2,1), (2,2).... (2,6), (4,1), (4,2).....(4,6), (6,1), (6,2)..... (6,6), (1,1), (1,2).... (1,6), (3,1), (3,2)..... (3,6), (5,1), (5,2).... (5,6) } = S
Hence, A and B are exhaustive.
TRUE
Question:7.(iii) Refer to question 6 above, state true or false: (give reason for your answer)
Answer:
Here,
S = {(x,y): 1 x,y 6}
A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
(iii) Therefore, B' = S -B = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} = A
TRUE
Question:7.(iv) Refer to question 6 above, state true or false: (give reason for your answer)
A and C are mutually exclusive
Answer:
Here,
S = {(x,y): 1 x,y 6}
A = { (2, 1), (2,2), (2,3) , (2,4), (2,5), (2,6), (4,1) , (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
C = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3) , (3,1), (3,2), (4,1) }
(iv) X and Y are mutually exclusive if and only if X Y =
A C = { (2,1), (2,2), (2,3) , (4,1) } ,
Hence, A and B are not mutually exclusive. FALSE
Question: 7.(v) Refer to question 6 above, state true or false: (give reason for your answer)
Answer:
X and Y are mutually exclusive if and only if X Y =
A B' = A A = A (From (iii))
A B’
Hence A and B' not mutually exclusive. FALSE
Question:7.(vi) Refer to question 6 above, state true or false: (give reason for your answer)
are mutually exclusive and exhaustive.
Answer:
Here,
S = {(x,y): 1 x,y 6}
A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}
C = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}
(vi) X and Y are mutually exclusive if and only if X Y =
A' B' = B A = (from (iii) and (i))
Hence A' and B' are mutually exclusive.
Again,
B' C = A C (from (iv))
Hence B' and C are not mutually exclusive.
Hence, A', B' and C are not mutually exclusive and exhaustive. FALSE
Class 11 maths chapter 16 NCERT solutions - Exercise: 16.3
Question:1(a) Which of the following can not be valid assignment of probabilities for outcomes of sample Space S
Answer:
(a) Condition (i): Each of the number p( ) is positive and less than one.
Condition (ii): Sum of probabilities = 0.1 + 0.01 + 0.05 + 0.03 + 0.01 + 0.2 + 0.6 = 1
Therefore, the assignment is valid
Question:1.(b) Which of the following cannot be the valid assignment of probabilities for outcomes of sample Space
Answer:
(b) Condition (i): Each of the number p( ) is positive and less than one.
Condition (ii): Sum of probabilities =
Therefore, the assignment is valid
Question:1.(c) Which of the following can not be valid assignment of probabilities for outcomes of sample Space
Answer:
(c) Since sum of probabilities = 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 = 2.8 > 1
Hence, Condition (ii) is not satisfied.
Therefore, the assignment is not valid
Question:1.(d) Which of the following can not be valid assignment of probabilities for outcomes of sample Space
Answer:
(d) Two of the probabilities p( ) and p( ) are negative, hence condition(i) is not satisfied.
Therefore, the assignment is not valid.
Question:1.(e) Which of the following can not be valid assignment of probabilities for outcomes of sample Space
Answer:
(e) Each of the number p( ) is positive but p( ) is not less than one. Hence the condition is not satisfied.
Therefore, the assignment is not valid.
Question:2 A coin is tossed twice, what is the probability that atleast one tail occurs?
Answer:
Sample space when a coin is tossed twice, S = {HH, HT, TH, TT}
[Note: A coin tossed twice is same as two coins tossed at once]
Number of possible outcomes n(S) = 4
Let E be the event of getting at least one tail = {HT, TH, TT}
n(E) = 3
= 0.75
Question:3.(i) A die is thrown, find the probability of following events:
Answer:
Sample space when a die is thrown, S = {1,2,3,4,5,6}
Number of possible outcomes n(S) = 6
Let E be the event of getting a prime number = {2,3,5}
n(E) = 3
= 0.5
Question:3.(ii) A die is thrown, find the probability of following events:
A number greater than or equal to will appear
Answer:
Sample space when a die is thrown, S = {1,2,3,4,5,6}
Number of possible outcomes n(S) = 6
Let E be the event of getting a number greater than or equal to 3 = {3,4,5,6}
n(E) = 4
= 0.67
Question:3.(iii) A die is thrown, find the probability of following events:
A number less than or equal to one will appear
Answer:
Sample space when a die is thrown, S = {1,2,3,4,5,6}
Number of possible outcomes n(S) = 6
Let E be the event of getting a number less than or equal to one = {1}
n(E) = 1
= 0.167
Question:3.(iv) A die is thrown, find the probability of following events:
A number more than will appear
Answer:
Sample space when a die is thrown, S = {1,2,3,4,5,6}
Number of possible outcomes n(S) = 6
Let E be the event of getting a number more than 6 will appear =
n(E) = 0
= 0
Question:3.(v) A die is thrown, find the probability of following events:
A number less than will appear.
Answer:
Sample space when a die is thrown, S = {1,2,3,4,5,6}
Number of possible outcomes n(S) = 6
Let E be the event of getting a number less than 6 will appear = {1,2,3,4,5}
n(E) = 5
= 0.83
Question:4(a). A card is selected from a pack of cards.
How many points are there in the sample space?
Answer:
(a) Number of points(events) in the sample space = Number of cards in the pack = 52
Question:4(b). A card is selected from a pack of cards.
Calculate the probability that the card is an ace of spades.
Answer:
Number of possible outcomes, n(S) = 52
Let E be the event that the card is an ace of spades
n(E) = 1
The required probability that the card is an ace of spades is .
Question:4(c)(i) A card is selected from a pack of 52 cards.
Calculate the probability that the card is an ace
Answer:
Number of possible outcomes, n(S) = 52
Let E be the event that the card is an ace. There are 4 aces.
n(E) = 4
The required probability that the card is an ace is .
Question:4(c)(ii) A card is selected from a pack of cards.
Calculate the probability that the card is black card.
Answer:
Number of possible outcomes, n(S) = 52
Let E be the event that the card is a black card. There are 26 black cards. (Diamonds and Clubs)
n(E) = 26
The required probability that the card is an ace is .
Answer:
The coin and die are tossed together.
The coin can have only 1 or 6 as possible outcomes and the die can have {1,2,3,4,5,6} as poosible outcomes
Sample space, S = {(x,y): x {1,6} and y {1,2,3,4,5,6}}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
Number of possible outcomes, n(S) = 12
(i) Let E be the event having sum of numbers as 3 = {(1, 2)}
n(E) = 1
The required probability of having 3 as sum of numbers is .
Answer:
The coin and die are tossed together.
The coin can have only 1 or 6 as possible outcomes and the die can have {1,2,3,4,5,6} as poosible outcomes
Sample space, S = {(x,y): x {1,6} and y {1,2,3,4,5,6}}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
Number of possible outcomes, n(S) = 12
(ii) Let E be the event having sum of numbers as 12 = {(6, 6)}
n(E) = 1
The required probability of having 12 as sum of numbers is .
Answer:
There are four men and six women on the city council
n(S) = n(men) + n(women) = 4 + 6 = 10
Let E be the event of selecting a woman
n(E) = 6
Therefore, the required probability of selecting a woman is 0.6
Answer:
Here the sample space is,
S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, THHT, HTTH, THTH, TTHH, TTTH, TTHT, THTT, HTTT, TTTT}
According to question,
1.) 4 heads = 1 + 1 + 1 + 1 = Rs. 4
2.) 3 heads and 1 tail = 1 + 1 + 1 - 1.50 = Rs. 1.50
3.) 2 heads and 2 tails = 1 + 1 - 1.50 - 1.50 = - Rs. 1 : he will lose Re. 1
4.) 1 head and 3 tails = 1 – 1.50 – 1.50 – 1.50 = - Rs. 3.50 : he will lose Rs. 3.50
5.) 4 tails = – 1.50 – 1.50 – 1.50 – 1.50 = - Rs. 6 = he will lose Rs. 6
Now, sample space of amounts corresponding to S:
S' = {4, 1.50, 1.50, 1.50, 1.50, - 1, - 1, - 1, - 1, - 1, - 1, - 3.50, - 3.50, - 3.50, - 3.50, - 6}
n(S') = 12
Required Probabilities are:
Question:8.(i) Three coins are tossed once. Find the probability of getting
heads
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting 3 heads = {HHH}
n(E) = 1
The required probability of getting 3 heads is .
Question 8.(ii) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting 2 heads = {HHT, HTH, THH}
n(E) = 3
The required probability of getting 2 heads is .
Question:âââââââ8.(iii) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting atleast 2 heads = Event of getting 2 or more heads = {HHH, HHT, HTH, THH}
n(E) = 4
The required probability of getting atleast 2 heads is .
Question:8.(iv) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting atmost 2 heads = Event of getting 2 or less heads = {HHT, HTH, THH, TTH, HTT, THT}
n(E) = 6
The required probability of getting almost 2 heads is .
Question:8.(v) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting no head = Event of getting only tails = {TTT}
n(E) = 1
The required probability of getting no head is .
Question:8.(vi) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting 3 tails = {TTT}
n(E) = 1
The required probability of getting 3 tails is .
Question:8(vii) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting exactly 2 tails = {TTH, HTT, THT}
n(E) = 3
The required probability of getting exactly 2 tails is .
Question:8.(viii) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting no tail = Event of getting only heads = {HHH}
n(E) = 1
The required probability of getting no tail is .
Question:8.(ix) Three coins are tossed once. Find the probability of getting
Answer:
Sample space when three coins are tossed: [Same as a coin tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Number of possible outcomes, n(S) = 8 [Note: 2x2x2 = 8]
Let E be the event of getting atmost 2 tails = Event of getting 2 or less tails = {HHT, HTH, THH, TTH, HTT, THT}
n(E) = 6
The required probability of getting atmost 2 tails is .
Question:9 If is the probability of an event, what is the probability of the event ‘not A’.
Answer:
Given,
P(E) =
We know,
P(not E) = P(E') = 1 - P(E)
=
=
Question: 10.(i) A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is a vowel
Answer:
Given, ‘ASSASSINATION’
No. of A's = 3; No. of S's = 4; No. of I's = 2; No. of N's = 2; No. of T = 1; No. of O = 1
No. of letters = 13
No. of vowels = {3 A's,2 I's,O} = 6
One letter is selected:
n(S) = = 13
Let E be the event of getting a vowel.
n(E) = = 6
Question: 10.(ii) A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is a consonant
Answer:
Given, ‘ASSASSINATION’
No. of A's = 3; No. of S's = 4; No. of I's = 2; No. of N's = 2; No. of T = 1; No. of O = 1
No. of letters = 13
No. of consonants = {4 S's,2 N's,T} = 7
One letter is selected:
n(S) = = 13
Let E be the event of getting a consonant.
n(E) = = 7
Answer:
Total numbers of numbers in the draw = 20
Numbers to be selected = 6
n(S) =
Let E be the event that six numbers match with the six numbers fixed by the lottery committee.
n(E) = 1 (Since only one prize to be won.)
Probability of winning =
Question:12.(i) Check whether the following probabilities and are consistently defined:
Answer:
(i) Given,
Now P(A B) > P(A)
(Since A B is a subset of A, P( A B) cannot be more than P(A))
Therefore, the given probabilities are not consistently defined.
Question:12.(ii) Check whether the following probabilities and are consistently defined
Answer:
(ii) Given,
We know,
P(A B) = P(A)+ P(B) - P(A B)
0.8 = 0.5 + 0.4 - P(A B)
P(A B) = 0.9 - 0.8 = 0.1
Therefore, P(A B) < P(A) and P(A B) < P(B) , which satisfies the condition.
Hence, the probabilities are consistently defined
Question:13 Fill in the blanks in following table:
Answer:
We know,
(i) = =
(ii)
(iii)
(i) | ||||
(ii) | 0.5 | |||
(iii) | 0.15 |
Question:14 Given and . Find , if and are mutually exclusive events.
Answer:
Given, and
To find :
We know,
[Since A and B are mutually exclusive events.]
Therefore,
Question:15(i) If E and F are events such that , and , find (i) P(E or F)
Answer:
Given, , and
To find :
We know,
Therefore,
Question:15.(ii) If E and F are events such that , and find P(not E and not F).
Answer:
Given, , and
To find :
We know,
And
Therefore,
Question:16 Events E and F are such that P(not E or not F) , State whether E and F are mutually exclusive.
Answer:
Given,
For A and B to be mutually exclusive,
Now,
We know,
Hence, E and F are not mutually exclusive.
Question:17(i) A and B are events such that P(A) , P(B) and P(A and B) . Determine (i) P(not A)
Answer:
Given, P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16
(i)
Therefore, P(not A) = 0.58
Question:17.(ii) A and B are events such that P(A) , P(B) and P(A and B) . Determine P(not B)
Answer:
Given, P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16
(ii)
Therefore, P(not B) = 0.52
Question:âââââââ17(iii) A and B are events such that P(A) , P(B) and P(A and B) . Determine P(A or B)
Answer:
Given, P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16
(iii) We know,
= 0.74
Answer:
Let M denote the event that the student is studying Mathematics and B denote the event that the student is studying Biology
And total students in the class be 100.
Given, n(M) = 40 P(M) =
n(B) = 30 P(M) =
n(M B) = 10 P(M) =
We know,
P(A B) = P(A)+ P(B) - P(A B)
P(M B) = 0.4 + 0.3 - 0.1 = 0.6
Hence, the probability that he will be studying Mathematics or Biology is 0.6
Answer:
Let A be the event that the student passes the first examination and B be the event that the students passes the second examination.
P(A B) is probability of passing at least one of the examination.
Therefore,
P(A B) = 0.95 , P(A)=0.8, P(B)=0.7
We know,
P(A B) = P(A)+ P(B) - P(A B)
P(A B) = 0.8 + 0.7 - 0.95 = 1.5 -0.95 = 0.55
Hence,the probability that the student will pass both the examinations is 0.55
Answer:
Let A be the event that the student passes English examination and B be the event that the students pass Hindi examination.
Given,
P(A)=0.75, P(A B) = 0.5, P(A' B') =0.1
We know,
P(A' B') = 1 - P(A B)
P(A B) = 1 - 0.1 = 0.9
Also,
P(A B) = P(A)+ P(B) - P(A B)
P(B) = 0.9 - 0.75 + 0.5 = 0.65
Hence,the probability of passing the Hindi examination is 0.65
Question:âââââââ21.(i) In a class of students, opted for NCC, opted for NSS and opted for both NCC and NSS. If one of these students is selected at random, find the probability that
The student opted for NCC or NSS.
Answer:
Let A be the event that student opted for NCC and B be the event that the student opted for NSS.
Given,
n(S) = 60, n(A) = 30, n(B) =32, n(A B) = 24
Therefore, P(A) =
P(B) =
P(A B) =
(i) We know,
P(A B) = P(A)+ P(B) - P(A B)
Hence,the probability that the student opted for NCC or NSS is
The student has opted neither NCC nor NSS.
Answer:
Let A be the event that student opted for NCC and B be the event that the student opted for NSS.
Given,
n(S) = 60, n(A) = 30, n(B) =32, n(A B) = 24
Therefore, P(A) =
P(B) =
P(A B) =
(ii) Now,
Probability that the student has opted neither NCC nor NSS = P(A' B' )
We know,
P(A' B' ) = 1 - P(A B) [De morgan's law]
And, P(A B) = P(A)+ P(B) - P(A B)
P(A' B' )
Hence,the probability that the student has opted neither NCC nor NSS. is
Answer:
Let A be the event that student opted for NCC and B be the event that the student opted for NSS.
Given,
n(S) = 60, n(A) = 30, n(B) =32, n(A B) = 24
Therefore, P(A) =
P(B) =
P(A B) =
(iii) Now,
Probability that the student has opted NSS but not NCC = P(B A' ) = P(B-A)
We know,
P(B-A) = P(B) - P(A B)
Hence,the probability that the student has opted NSS but not NCC is
Class 11 Probability NCERT solutions - Miscellaneous Exercise
Answer:
Given,
No. of red marbles = 10
No. of blue marbles = 20
No. of green marbles = 30
Total number of marbles = 10 + 20 + 30 = 60
Number of ways of drawing 5 marbles from 60 marbles =
(i) .
Number of ways of drawing 5 blue marbles from 20 blue marbles =
Probability of drawing all blue marbles =
Question:1.(ii) A box contains red marbles, blue marbles and green marbles. marbles are drawn from the box, what is the probability that atleast one will be green?
Answer:
Given,
No. of red marbles = 10
No. of blue marbles = 20
No. of green marbles = 30
Total number of marbles = 10 + 20 + 30 = 60
Number of ways of drawing 5 marbles from 60 marbles =
(ii). We know,
The probability that at least one marble is green = 1 - Probability that no marble is green
Now, Number of ways of drawing no green marbles = Number of ways of drawing only red and blue marbles
=
The probability that no marble is green =
The probability that at least one marble is green =
Answer:
Total number of ways of drawing 4 cards from a deck of 52 cards =
We know that there are 13 diamonds and 13 spades in a deck.
Now, Number of ways of drawing 3 diamonds and 1 spade =
Probability of obtaining 3 diamonds and 1 spade
Answer:
Total number of faces of a die = 6
(i) Number of faces with number '2' = 3
Therefore, required probability P(2) is 0.5
Question:3.(ii) A die has two faces each with number ‘1’, three faces each with number ‘2’ and one face with number ‘3’. If die is rolled once, determine
Answer:
Total number of faces of a die = 6
(ii) P (1 or 3) = P (not 2) = 1 − P (2)
Therefore, required probability P(1 or 3) is 0.5
Question:3.(iii) A die has two faces each with number ‘1’, three faces each with number ‘2’ and one face with number ‘3’. If die is rolled once, determine
Answer:
Total number of faces of a die = 6
(iii) Number of faces with number '3' = 1
P(not 3) = 1 - P(3)
Therefore, required probability P(not 3) is
Answer:
Given, Total number of tickets sold = 10,000
Number of prizes awarded = 10
(a) If one ticket is bought,
P(getting a prize) =
P(not getting a prize) = 1 - P(getting a prize)
Answer:
Given, Total number of tickets sold = 10,000
Number of prizes awarded = 10
(b) If two tickets are bought:
Number of tickets not awarded = 10000 - 10 = 9990
P(not getting a prize) =
Answer:
Given, Total number of tickets sold = 10,000
Number of prizes awarded = 10
(c) If ten tickets are bought:
Number of tickets not awarded = 10000 - 10 = 9990
P(not getting a prize) =
Question:5.(a) Out of students, two sections of and are formed. If you and your friend are among the students, what is the probability that
you both enter the same section?
Answer:
Total number of students = 100
Let A and B be the two sections consisting of 40 and 60 students respectively.
Number of ways of selecting 2 students out of 100 students.=
If both are in section A:
Number of ways of selecting 40 students out of 100 = (The remaining 60 will automatically be in section B!)
Remaining 38 students are to be chosen out of (100-2 =) 98 students
Required probability if they both are in section A =
Similarly,
If both are in section B:
Number of ways of selecting 60 students out of 100 = (The remaining 40 will automatically be in section A!)
Remaining 58 students are to be chosen out of (100-2 =) 98 students
Required probability if they both are in section B =
Required probability that both are in same section = Probability that both are in section A + Probability that both are in section B
=
Hence, the required probability that both are in same section is
Question:5.(b) Out of students, two sections of and are formed. If you and your friend are among the students, what is the probability that
you both enter the different sections?
Answer:
Total number of students = 100
Let A and B be the two sections consisting of 40 and 60 students respectively.
We found out in (a) that the probability that both students are in same section is
(b) Probability that both the students are in different section =
Answer:
Given, 3 letters are put in 3 envelopes.
The number of ways of putting the 3 different letters randomly = 3!
Number of ways that at least one of the 3 letters is in the correct envelope
= No. of ways that exactly 1 letter is in correct envelope + No. of ways that 2 letters are in the correct envelope(The third is automatically placed correctly)
= No. of ways that exactly 1 letter is in correct envelope + No. of ways that all the 3 letters are in the correct envelope
=
(Explanation for :
No. of ways of selecting 1 envelope out of 3 = .
If we put the correct letter in it, there is only one way the other two are put in the wrong envelope! )
Therefore, the probability that at least one letter is in its proper envelope =
Question:7(i) A and B are two events such that , and Find
Answer:
Given, P(A) = 0.54, P(B) = 0.69, P(A B) = 0.35
(i) We know, P(A B) = P(A) + P(B) - P(A B)
P(A B) = 0.54 + 0.69 - 0.35 = 0.88
P(A B) = 0.88
Question:7(ii) A and B are two events such that , and . Find
Answer:
Given,
And,
(ii) [De Morgan’s law]
So,
Question:7(iii) A and B are two events such that , and Find
Answer:
Given, P(A) = 0.54, P(B) = 0.69, P(A B) = 0.35
And, P(A B) = 0.88
(iii) P(A B') = P(A-B) = P(A) - P(A B)
= 0.54 - 0.35 = 0.19
P(A B') = 0.19
Question:7(iv) A and B are two events such that , and . Find
Answer:
Given, P(A) = 0.54, P(B) = 0.69, P(A B) = 0.35
And, P(A B) = 0.88
(iv) P(B A') = P(B-A) = P(B) - P(A B)
= 0.69 - 0.35 = 0.34
P(B A') = 0.34
S. No. | Name | Sex | Age in years |
1. | Harish | M | 30 |
2. | Rohan | M | 33 |
3. | Sheetal | F | 46 |
4. | Alis | F | 28 |
5. | Salim | M | 41 |
A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over years?
Answer:
Given,
Total number of persons = 5
No. of male spokesperson = 3
No. of spokesperson who is over 35 years of age = 2
Let E be the event that the spokesperson is a male and F be the event that the spokesperson is over 35 years of age.
Since only one male is over 35 years of age,
We know,
Therefore, the probability that the spokesperson will either be a male or over 35 years of age is .
Answer:
Since 4-digit numbers greater than 5000 are to be formed,
The place digit can be filled up by either 7 or 5 in ways
Since repetition is allowed,
Each of the remaining 3 places can be filled by any of the digits 0, 1, 3, 5, or 7 in ways.
Total number of 4-digit numbers greater than 5000 =
(5000 cannot be counted, hence one less)
We know, a number is divisible by 5 if unit’s place digit is either 0 or 5.
Total number of 4-digit numbers greater than 5000 that are divisible by 5 =
Therefore, the required probability =
Answer:
Since 4-digit numbers greater than 5000 are to be formed,
The place digit can be filled up by either 7 or 5 in ways
Since repetition is not allowed,
The remaining 3 places can be filled by remaining 4 digits in ways.
Total number of 4-digit numbers greater than 5000 =
We know, a number is divisible by 5 if unit’s place digit is either 0 or 5.
Case 1. When digit at place is 5, the units place can be filled only with 0.
And the & places can be filled with any two of the remaining digits {1,3,7} in
Number of 4-digit numbers starting with 5 and divisible by 5 =
Case 2. When digit at place is 7, the units place can be filled by 0 or 5 in 2 ways.
And the & places can be filled with any two of the remaining 3 digits in
Number of 4-digit numbers starting with 7 and divisible by 5 =
Total number of 4-digit numbers greater than 5000 that are divisible by 5 = 6 + 12 = 18
Therefore, the required probability =
Answer:
Given, Each wheel can be labelled with 10 digits.
Number of ways of selecting 4 different digits out of the 10 digits =
These 4 digits can arranged among themselves is ways.
Number of four digit numbers without repetitions =
Number of combination that can open the suitcase = 1
Required probability of getting the right sequence to open the suitcase =
Section 16.1 - Introduction
This section presents a concise explanation of the concept of probability and its associated terminology. In basic terms, probability refers to the proportion of favorable outcomes to the total number of possible outcomes. Other relevant terms such as experiment, sample space, events, and outcomes are also discussed.
Section 16.2 – Random Experiments
This section, 16.2.1, focuses on outcomes and sample space in random experiments. The definition of a random experiment, along with the necessary conditions and relevant terms, is elaborated upon with the help of examples such as coin tossing and rolling dice. This approach aims to facilitate a clear understanding of the topic for students.
Section 16.3 – Event
The following sub-topics have been covered in this section, 16.3:
Numerous examples and experiments have been used to elucidate these topics and provide a clear understanding of the related terms and event types. The algebra of events is particularly helpful in comprehending concepts such as intersection, union, difference, and complement of events.
Section 16.4 – Axiomatic Approach to Probability
This section, 16.4, covers various methods of calculating probabilities. The sub-topics are as follows:
The axiomatic approach is another way to define probability, and this is comprehensively explained in Class 11 Maths Chapter 16 of the NCERT curriculum. Furthermore, numerous examples and practice problems are available to aid in the comprehension of these concepts.
Intrested students can practice class 11 maths ch 16 question answer using the following exercises.
chapter-1 | |
chapter-2 | |
chapter-3 | |
chapter-4 | |
chapter-5 | |
chapter-6 | |
chapter-7 | |
chapter-8 | |
chapter-9 | |
chapter-10 | |
chapter-11 | |
chapter-12 | |
chapter-13 | |
chapter-14 | |
chapter-15 | |
chapter-16 |
Some key features of ncert solutions for class 11 maths chapter 16 :
Comprehensive Coverage: The probability class 11 solutions provide a detailed explanation of all the key concepts of probability, including sample space, events, probability distribution, conditional probability, and more.
Easy-to-understand language: The ch 16 maths class 11 solutions are written in simple language, making it easy for students to comprehend and learn the concepts without any difficulty.
Chapter-wise Exercise Solutions: The class 11 probability NCERT solutions provide chapter-wise exercise solutions, which include both short answer and long answer type questions, along with step-by-step solutions.
The ch 16 maths class 11 includes topics such as Basic probability problems, Random Experiments, Events, and Axiomatic Approach to Probability. students can practice these topics from NCERT solutions as discussed above in this article.
The concept of Conditional Probability in chapter 16 class 11 maths refers to the situation where the occurrence of one event is dependent on the occurrence of another event. In this case, both events are considered to be independent of each other, meaning that the probability of event A does not affect the probability of event B. The solutions provide a detailed explanation of this concept, which can help students score well in their Class 11 exam.
In the NCERT Solutions for Class 11 Maths Chapter 16, the concept of sample space refers to a set of all possible outcomes of a future event, such as picking a card from a deck of cards. This set is defined by the method used to select an item from a given sample, and every probability is a subset of the sample space. It is important to note that the sum of all probabilities within a sample space is always equal to one. The solutions for this chapter, available in PDF format, provide a detailed explanation of this concept to help students better understand it.
there are major topics and sub topics in class 11th maths chapter 16:
Here you will get the detailed NCERT solutions for class 11 maths by clicking on the link. students can these solutions to command and indepth understanding the concepts. this will help students to build confidence overall lead to score well in the exams. for ease they can study probability class 11 pdf both online and offline mode.
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For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.
An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.
They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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