
CBSE Important Questions›

CBSE Previous Year Question Papers›
 CBSE Previous Year Question Papers
 CBSE Previous Year Question Papers Class 12
 CBSE Previous Year Question Papers Class 10

CBSE Revision Notes›

CBSE Syllabus›

CBSE Extra Questions›

CBSE Sample Papers›
 CBSE Sample Papers
 CBSE Sample Question Papers For Class 5
 CBSE Sample Question Papers For Class 4
 CBSE Sample Question Papers For Class 3
 CBSE Sample Question Papers For Class 2
 CBSE Sample Question Papers For Class 1
 CBSE Sample Question Papers For Class 12
 CBSE Sample Question Papers For Class 11
 CBSE Sample Question Papers For Class 10
 CBSE Sample Question Papers For Class 9
 CBSE Sample Question Papers For Class 8
 CBSE Sample Question Papers For Class 7
 CBSE Sample Question Papers For Class 6

ISC & ICSE Syllabus›

ICSE Question Paper›
 ICSE Question Paper
 ISC Class 12 Question Paper
 ICSE Class 10 Question Paper

ICSE Sample Question Papers›
 ICSE Sample Question Papers
 ISC Sample Question Papers For Class 12
 ISC Sample Question Papers For Class 11
 ICSE Sample Question Papers For Class 10
 ICSE Sample Question Papers For Class 9
 ICSE Sample Question Papers For Class 8
 ICSE Sample Question Papers For Class 7
 ICSE Sample Question Papers For Class 6

ICSE Revision Notes›
 ICSE Revision Notes
 ICSE Class 9 Revision Notes
 ICSE Class 10 Revision Notes

ICSE Important Questions›

Maharashtra board›

RajasthanBoard›
 RajasthanBoard

Andhrapradesh Board›
 Andhrapradesh Board
 AP Board Sample Question Paper
 AP Board syllabus
 AP Board Previous Year Question Paper

Telangana Board›

Tamilnadu Board›

NCERT Solutions Class 12›
 NCERT Solutions Class 12
 NCERT Solutions Class 12 Economics
 NCERT Solutions Class 12 English
 NCERT Solutions Class 12 Hindi
 NCERT Solutions Class 12 Maths
 NCERT Solutions Class 12 Physics
 NCERT Solutions Class 12 Accountancy
 NCERT Solutions Class 12 Biology
 NCERT Solutions Class 12 Chemistry
 NCERT Solutions Class 12 Commerce

NCERT Solutions Class 10›

NCERT Solutions Class 11›
 NCERT Solutions Class 11
 NCERT Solutions Class 11 Statistics
 NCERT Solutions Class 11 Accountancy
 NCERT Solutions Class 11 Biology
 NCERT Solutions Class 11 Chemistry
 NCERT Solutions Class 11 Commerce
 NCERT Solutions Class 11 English
 NCERT Solutions Class 11 Hindi
 NCERT Solutions Class 11 Maths
 NCERT Solutions Class 11 Physics

NCERT Solutions Class 9›

NCERT Solutions Class 8›

NCERT Solutions Class 7›

NCERT Solutions Class 6›

NCERT Solutions Class 5›
 NCERT Solutions Class 5
 NCERT Solutions Class 5 EVS
 NCERT Solutions Class 5 English
 NCERT Solutions Class 5 Maths

NCERT Solutions Class 4›

NCERT Solutions Class 3›

NCERT Solutions Class 2›
 NCERT Solutions Class 2
 NCERT Solutions Class 2 Hindi
 NCERT Solutions Class 2 Maths
 NCERT Solutions Class 2 English

NCERT Solutions Class 1›
 NCERT Solutions Class 1
 NCERT Solutions Class 1 English
 NCERT Solutions Class 1 Hindi
 NCERT Solutions Class 1 Maths

JEE Main Question Papers›

JEE Main Syllabus›
 JEE Main Syllabus
 JEE Main Chemistry Syllabus
 JEE Main Maths Syllabus
 JEE Main Physics Syllabus

JEE Main Questions›
 JEE Main Questions
 JEE Main Maths Questions
 JEE Main Physics Questions
 JEE Main Chemistry Questions

JEE Main Mock Test›
 JEE Main Mock Test

JEE Main Revision Notes›
 JEE Main Revision Notes

JEE Main Sample Papers›
 JEE Main Sample Papers

JEE Advanced Question Papers›

JEE Advanced Syllabus›
 JEE Advanced Syllabus

JEE Advanced Mock Test›
 JEE Advanced Mock Test

JEE Advanced Questions›
 JEE Advanced Questions
 JEE Advanced Chemistry Questions
 JEE Advanced Maths Questions
 JEE Advanced Physics Questions

JEE Advanced Sample Papers›
 JEE Advanced Sample Papers

NEET Eligibility Criteria›
 NEET Eligibility Criteria

NEET Question Papers›

NEET Sample Papers›
 NEET Sample Papers

NEET Syllabus›

NEET Mock Test›
 NEET Mock Test

NCERT Books Class 9›
 NCERT Books Class 9

NCERT Books Class 8›
 NCERT Books Class 8

NCERT Books Class 7›
 NCERT Books Class 7

NCERT Books Class 6›
 NCERT Books Class 6

NCERT Books Class 5›
 NCERT Books Class 5

NCERT Books Class 4›
 NCERT Books Class 4

NCERT Books Class 3›
 NCERT Books Class 3

NCERT Books Class 2›
 NCERT Books Class 2

NCERT Books Class 1›
 NCERT Books Class 1

NCERT Books Class 12›
 NCERT Books Class 12

NCERT Books Class 11›
 NCERT Books Class 11

NCERT Books Class 10›
 NCERT Books Class 10

Chemistry Full Forms›
 Chemistry Full Forms

Biology Full Forms›
 Biology Full Forms

Physics Full Forms›
 Physics Full Forms

Educational Full Form›
 Educational Full Form

Examination Full Forms›
 Examination Full Forms

Algebra Formulas›
 Algebra Formulas

Chemistry Formulas›
 Chemistry Formulas

Geometry Formulas›
 Geometry Formulas

Math Formulas›
 Math Formulas

Physics Formulas›
 Physics Formulas

Trigonometry Formulas›
 Trigonometry Formulas

CUET Admit Card›
 CUET Admit Card

CUET Application Form›
 CUET Application Form

CUET Counselling›
 CUET Counselling

CUET Cutoff›
 CUET Cutoff

CUET Previous Year Question Papers›
 CUET Previous Year Question Papers

CUET Results›
 CUET Results

CUET Sample Papers›
 CUET Sample Papers

CUET Syllabus›
 CUET Syllabus

CUET Eligibility Criteria›
 CUET Eligibility Criteria

CUET Exam Centers›
 CUET Exam Centers

CUET Exam Dates›
 CUET Exam Dates

CUET Exam Pattern›
 CUET Exam Pattern
Important Questions Class 11 Mathematics Chapter 16
Important Questions for CBSE Class 11 Mathematics Chapter 16 – Probability
Probability is a measure of how likely an event is to occur. It is mathematically defined as the ratio of the number of favourable outcomes of an event to the total number of outcomes of the event.
Quick Links
ToggleThe formula for calculating the probability of an event is to divide the “x” number of favourable outcomes by the “n” total number of possible outcomes. It can be represented as follows.
Event Probability = Favourable Outcomes/Total Outcomes = x/n
The number of favourable outcomes cannot exceed the total number of outcomes.
We frequently use probability in our daily lives without giving it much thought. Similarly, when it comes to using an informal method of probability, researchers take their time determining the accurate results of probability to assist healthcare providers, insurance companies, stakeholders, and many other critical decisions.
Probability can vary depending on how you approach the problem. When considering outcomes over a long period of repetition, the answer is 50:50. This demonstrates an important aspect of how probability works.
The topics that are covered in this chapter are as follows.
 Probability
 Introduction
 Random Experiments
 Event
 Axiomatic Approach to Probability
CBSE Class 11 Mathematics Chapter16 Important Questions
Study Important Questions for Class 11 Mathematics Chapter 16 – Probability
Extramarks Important Questions for Class 11 Mathematics Chapter 16 is useful for students who want to study the chapter in a questionanswer format. These questions are compiled by subject matter experts from NCERT books and past year’s question papers. These are written in a clear, concise manner so that the students understand how to write answers for examinations and learn the topic at hand.
Given below are Important Questions for 1, 4, and 6 marks each. To access the complete set of questions, click the link for the Extramarks website.
Very Short Answers and Questions
1 Mark
Q1. What are the odds that a leap year, chosen at random, will have 53 Sundays?
A1. The total number of days in a leap year is 366, with 52 complete weeks and two days extra. The two days could be (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), or (Sunday, Monday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday, Tuesday).
P(53 Sundays in a leap year) = 2/7
Q2. Describe the test space if a coin is tossed, followed by a die.
A2. H1, H2, H3, H4, H5, T1, T2, T3, T4, T5, T6
Where, H Heads
T Tails
Q3. We would like to select one child from a group of two boys and three girls. A coin is tossed, and if it comes up heads, a boy is selected; otherwise, a girl is selected. Describe the test space.
A3. HB1, HB2, TG1, TG2, TG3 are the answers.
Where, HB Heads Boy
TG Tails Girl
Q4. A box contains one white ball and three identical black balls. Two balls are drawn at random, one after the other, with no replacement. Create the experiment’s sample space.
A4. S = {WB, BW, and BB.}
Where, WB White Ball
BW Black white
BB Black Ball
Long Answers and Questions
4 Marks
Q1. Consider the following scenario: A coin is tossed three times. They form a set of mutually exclusive and exhaustive events if A: no head appears, B: exactly one head appears, and C: at least two heads appear.
A1. Let H represent the outcome of heads every time the coin is tossed.
Let T represent the outcome of tails every time the coin is tossed.
The sample size S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
From the given data, A = {TTT}, B={HTT, THT, TTH}, C = {HHT, HTH, THH, HHH}
A∪B∪C = S
Therefore, A, B, and C are exhaustive events.
Also, A∩B = ∅, A∩C = ∅, C∩C=∅, disjoint, i.e., they are mutually exclusive.
Q2. 40% of students in a school’s Class 11 study Mathematics, while 30% study Biology. 10% of the students in the class study both Mathematics and Biology. Determine whether a student will be studying Mathematics, Biology, or both subjects if selected randomly.
A2. Let P(M) represent the probability of students studying Mathematics.
Let P(B) represent the probability of students studying Biology.
P(M) = 40100, P(B) = 30100
P(M∩B) = 10100
P(M∪B) = P(M) + P(B) – P(M∩B)
= 40100 + 30100 – 10100 = 60100= 0.6
Q3. A hockey game is played from 3 to 5 p.m. Write the problem if a man arrives late for the game and misses the only goal of the game, which is scored in the 20th minute?
A3. Total time = 3 pm – 5 pm = 2hr = 120 min
He can see the goal only if he arrives within the initial 20 minutes.
P(he sees the goal) = 20/120=1/6
P(he does not see the goal) = 1−1/6=5/6
Q4. In a town of 6000 people, 1200 are over the age of 50, and 2000 are women. It is well known that 30% of females are over the age of 50. What is the likelihood that a randomly selected individual from the town is either female or over 30 years old?
A4. Let A1 be the event that the person is a female and A2 be the event that the person is 50 yr. old.
n(A1) = 2000,n(A2) = 1200
n(A1∩A2) = 30% of 2000 = 30/100×2000=600n(A1∪A2) = n(A1) + n(A2) – n(A1∩A2)
= 2000 + 120 – 600 = 2600
P(A1∪A2) = 2600/6000 = 1330
Q5. If an entrance exam is graded on the basis of two examinations, a randomly selected student has a 0.8 chance of passing the first examination and a 0.7 chance of passing the second examination. The chances of passing at least one of them are 0.95. What is the likelihood of passing both?
A5. Let A represent the event where the student passes the first examination
Let B represent the event where the student passes the second examination
P(A) = 0.8,P(B) = 0.7
P(A∪B) = 0.95
P(A∩B) = ?
P(A∪B) = P(A) + P(B) – P(A∩B)
0.95 = 0.8 + 0.7 – P(A∩B)
0.55 = P(A∩B)
Very Long Answers and Questions
6 Marks
Q1. Three letters are dictated to three people, and each of them receives an envelope. The letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Determine whether or not at least one letter is in its proper envelope.
A1. Assume the three letters are A1, A2, and A3, and the three envelopes are E1, E2, and E3.
3P3 = 6 ways to put letters in three envelopes
Methods in which none of the letters is properly enveloped = 2
Probability:
P(at least one letter placed in an appropriate envelope) = 1 – P (none letters is put into a proper envelope)
= 1−2/6
= 2/3
Q2. The cards are numbered 1 to 20. One card is drawn at random. What is the probability that the number on the drawn card is
(i) a prime number; (ii) an odd number; (iii) a multiple of five; and (iv) not divisible by three?
A2. Let the sample space be ‘S’.
S = {1, 2, 3, 4,……., 20}
Let the events E1, E2, E3, and E4 each have a prime number, an odd number, a multiple of 5, and are not divisible by 3.
P(E1) = 820 = 25,E1 = { 2,3,5,7,11,13,17,19}
P(E2) = 1020 = 12,E2 = { 1,3,5,7,9,11,13,15,17,19}
P(E3) = 420 = 15,E3 = { 5,10,15,20}
P(E4) = 1420 = 710,E4 = { 1,2,4,5,7,8,10,11,13,14,16,17,19,20}
Q.1 Ashok and Amit play a game, where, each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial is
Marks:1
Ans
The number of ways in which either of the players can choose from 1 to 25 is 25.
So, the total number of ways of choosing numbers = 25 25
= 625.
There are 25 ways in which the numbers chosen by both the players is the same.
Therefore, the probability that they will win a prize in a single trial = 25/625
= 1/25.
Hence, the probability that they will not win a prize in a single trial = 1 1/25
= 24/25.
Q.2 Two cards are drawn from a pack of 52 cards. The probability that either both are red or both are kings, is
55/221.
105/663.
125/663.
135/663.
Marks:1
Ans
55/221.
Exhaustive no. of cases = ^{52}C_{2} = 1326.
P(A) = Probability of red card
P(B) = Probability of king
Q.3 A, B and C are mutually exclusive and exhaustive events associated with a random experiment.
Find P(A) if P(B) = (5/2)P(A) and P(C) = (3/2)P(A).
Marks:1
Ans
Let P(A) = p
P(B) = (5/2) P(A)
= (5/2)P
and
P(C) = (3/2)P(A)
= (3/2)P
Since A, B and C are mutually exclusive and exhaustive events associated with a random experiment, thus A ª B ª C = S
P( A ª B ª C) = P(S) = 1
P(A) + P(B) + P(C) = 1
p + (5/2)p + (3/2)p = 1
(10/2)p =1
p = 1/5
P(A) = p
= 1/5.
Q.4 A and B are two students. Probabilities of solving a problem by A and B separately are 5/9 and 7/11 respectively. Find the probability that the problem will be solved.
Marks:4
Ans
Given that P(A) = 5/9 and (P(B) = 7/11
P(A^{c}) = 1 P(A)
= 4/9
P(B^{c}) = 1 P(B)
= 4/11
The problem will be solved if at least one of the two is able to solve the problem.
Therefore, Required Probability = P(A âˆª B)
= 1P(A^{c}). P(B^{c})
= 1(4/9)— (4/11)
= 83/99
Q.5 There are 3 letters and 3 addressed envelopes. Find the probability that all the letters are not dispatched in the right envelopes.
Marks:4
Ans
3 letters can be put in 3 addressed envelopes in 3 ways.
3 letters can be put in 3 addressed envelopes correctly in 1 way.
Probability that all the letters are dispatched in the right envelopes = 1/3 = 1/6
Probability that all the letters are not dispatched in the right envelopes = 1 (1/6) = 5/6.
Please register to view this section
FAQs (Frequently Asked Questions)
1. What is a sample space and how does it help?
A set is made up of all the possible outcomes of a future event, such as drawing a card from a deck of cards. It can be defined as the process of selecting something from a given sample and sample space. Every probability represents a small portion of the sample space. As a result, the total of all possible outcomes will be one. Furthermore, if you try to find the probability of an event occurring you will notice that it is always one minus the probability that it did not occur.
2. What do you mean by Conditional Probability?
A Conditional Probability is an event that occurs only after confirmation that the first event has occurred. In these cases, the events that will occur are unrelated to one another. As a result, event A has no effect on the probability of event B, making Event A simply the Probability of event B.
3. What is the significance of the Random Experiment?
A random experiment is a test, trial, or observation that can be repeated numerous times under identical conditions. The outcome of a single random experiment must be independent and uniformly distributed. It is an important aspect of the topic of Probability in Class 11 Mathematics Chapter 16.