Find the best tutors and institutes for Class 11 Tuition
Q2:
A coin is tossed twice, what is the probability that atleast one tail occurs?
Solution :
Step 1: Definition of the Sample Space ($S$)
When a fair coin is tossed, there are two possible outcomes: Heads ($H$) or Tails ($T$). For a compound experiment involving two independent tosses, the sample space $S$ is the Cartesian product of the individual outcomes: $\{H, T\} \times \{H, T\}$.
The total number of outcomes $n(S)$ is given by $2^n$, where $n=2$ tosses: $2^2 = 4$.
$S = \{(H, H), (H, T), (T, H), (T, T)\}$
[By the Fundamental Counting Principle, $2 \times 2 = 4$ total outcomes.]
Step 2: Defining the Favorable Event ($E$)
Let $E$ be the event of getting "at least one tail." This condition is satisfied if the outcome contains one tail or two tails.
- $(H, T)$: Contains one tail.
- $(T, H)$: Contains one tail.
- $(T, T)$: Contains two tails.
Thus, $E = \{(H, T), (T, H), (T, T)\}$.
The number of favorable outcomes $n(E) = 3$.
Step 3: Calculating Probability ($P(E)$)
The probability of an event $E$ is defined by the ratio of favorable outcomes to total possible outcomes in the sample space [Per the Classical Definition of Probability]:
$P(E) = \frac{n(E)}{n(S)}$
$P(E) = \frac{3}{4}$
Visual Representation of Sample Space
Alternative Method: Complementary Probability
Using the Complement Rule: $P(E) = 1 - P(E^c)$, where $E^c$ is the event "no tails occur" (i.e., only heads).
$E^c = \{(H, H)\}$, so $n(E^c) = 1$.
$P(E^c) = \frac{1}{4}$
$P(E) = 1 - \frac{1}{4} = \frac{3}{4}$
Final Solution: The probability of getting at least one tail is $\frac{3}{4}$ or $0.75$.
More Questions from Class 11 Mathematics Probability EXERCISE 14.2
- Q1: Which of the following can not be valid assignment of probabilities for outcomes of sample Space $S = \{\omega_1, \omega_2, \omega_3, \omega_4, \omega_5, \omega_6, \omega_7\}$
- Q3(i): A die is thrown, find the probability of following events: (i) A prime number will appear,
- Q3(ii): A die is thrown, find the probability of following events: (ii) A number greater than or equal to 3 will appear,
- Q3(iii): A die is thrown, find the probability of following events: (iii) A number less than or equal to one will appear,
- Q3(iv): A die is thrown, find the probability of following events: (iv) A number more than 6 will appear,
- Q3(v): A die is thrown, find the probability of following events: (v) A number less than 6 will appear.
- Q4(a): A card is selected from a pack of 52 cards. (a) How many points are there in the sample space?
- Q4(b): A card is selected from a pack of 52 cards. (b) Calculate the probability that the card is an ace of spades.
- Q4(c)(i): A card is selected from a pack of 52 cards. (c) Calculate the probability that the card is (i) an ace
- Q4(c)(ii): A card is selected from a pack of 52 cards. (c) Calculate the probability that the card is (ii) black card.
- Q5(i): 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
- Q5(ii): 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 (ii) 12
- Q6: 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?
- Q7: 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.
- Q8(i): Three coins are tossed once. Find the probability of getting (i) 3 heads
- Q8(ii): Three coins are tossed once. Find the probability of getting (ii) 2 heads
- Q8(iii): Three coins are tossed once. Find the probability of getting (iii) atleast 2 heads
- Q8(iv): Three coins are tossed once. Find the probability of getting (iv) atmost 2 heads
- Q8(ix): Three coins are tossed once. Find the probability of getting (ix) atmost two tails
- Q8(v): Three coins are tossed once. Find the probability of getting (v) no head
- Q8(vi): Three coins are tossed once. Find the probability of getting (vi) 3 tails
- Q8(vii): Three coins are tossed once. Find the probability of getting (vii) exactly two tails
- Q8(viii): Three coins are tossed once. Find the probability of getting (viii) no tail
- Q9: If $\frac{2}{11}$ is the probability of an event, what is the probability of the event ‘not A’.
- Q10(i): A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is (i) a vowel
- Q10(ii): A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is (ii) a consonant
- Q11: 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.]
- Q12(i): Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined (i) $P(A) = 0.5, P(B) = 0.7, P(A \cap B) = 0.6$
- Q12(ii): Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined (ii) $P(A) = 0.5, P(B) = 0.4, P(A \cup B) = 0.8$
- Q13(i): Fill in the blanks in following table: (i) $P(A) = \frac{1}{3}, P(B) = \frac{1}{5}, P(A \cap B) = \frac{1}{15}, P(A \cup B) = ...$
- Q13(ii): Fill in the blanks in following table: (ii) $P(A) = 0.35, P(B) = ..., P(A \cap B) = 0.25, P(A \cup B) = 0.6$
- Q13(iii): Fill in the blanks in following table: (iii) $P(A) = 0.5, P(B) = 0.35, P(A \cap B) = ..., P(A \cup B) = 0.7$
- Q14: Given $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$. Find $P(A \text{ or } B)$, if A and B are mutually exclusive events.
- Q15(i): If E and F are events such that $P(E) = \frac{1}{4}$, $P(F) = \frac{1}{2}$ and $P(E \text{ and } F) = \frac{1}{8}$, find (i) $P(E \text{ or } F)$,
- Q15(ii): If E and F are events such that $P(E) = \frac{1}{4}$, $P(F) = \frac{1}{2}$ and $P(E \text{ and } F) = \frac{1}{8}$, find (ii) $P(\text{not } E \text{ and not } F)$.
- Q16: Events E and F are such that $P(\text{not } E \text{ or not } F) = 0.25$, State whether E and F are mutually exclusive.
- Q17(i): A and B are events such that $P(A) = 0.42, P(B) = 0.48$ and $P(A \text{ and } B) = 0.16$. Determine (i) $P(\text{not } A)$,
- Q17(ii): A and B are events such that $P(A) = 0.42, P(B) = 0.48$ and $P(A \text{ and } B) = 0.16$. Determine (ii) $P(\text{not } B)$ and
- Q17(iii): A and B are events such that $P(A) = 0.42, P(B) = 0.48$ and $P(A \text{ and } B) = 0.16$. Determine (iii) $P(A \text{ or } B)$
- Q18: 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.
- Q19: 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 atleast one of them is 0.95. What is the probability of passing both?
- Q20: 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?
- Q21(i): 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.
- Q21(ii): 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 (ii) The student has opted neither NCC nor NSS.
- Q21(iii): 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 (iii) The student has opted NSS but not NCC.
CBSE Solutions for Class 11 Mathematics Probability
Chapters in CBSE - Class 11 Mathematics
Top Tutors who teach Probability
Top Tutor
Ramachandra Puram, Hyderabad
10 yrs of
Exp
4 students
243 hrs of classes completed
400
per hour
Online Classes
Student's Home
I am an experienced academic teacher and tutor, specialised in teaching Accountancy to Class 11 students. My teaching approach focuses on building strong fundamentals, simplifying complex chapters and explaining practical problems step by step and making learning engaging and effective. Through interactive teaching, personalised attention and exam-oriented practice, I help students prepare effectively and perform well in their exams.
He teaches really well so that we can understand the concept.His concept understanding is very good.
Top Tutor
Ashok Nagar Block 14, Delhi
17 yrs of
Exp
7 students
30 hrs of classes completed
800
per hour
Online Classes
Student's Home
Tutor's home
I am teacher with experience of 17 years. Provides notes and assignments.
Took online classes from Deepti maam . She is knowledgeable and professional teacher. Took regular test. Highly recommended to take online class as there is no difference in online and student home . Lectures taken from her.
Top Tutor
Kommadi, Visakhapatnam
8 yrs of
Exp
1 student
29 hrs of classes completed
800
per hour
Online Classes
I am Kirti from Vizag. I have around 8 years of teaching experience and have taught in reputed central schools like Kendriya Vidyalaya, IIT schools, and also international schools. Teaching should be fun according to me, not a burden to students. I teach students in a friendly way, connecting different examples of the world.
keerthi ma'am is the best teacher I've come across. She's precise with her teaching and has a remarkable teaching style, which is much easier to understand. She made me find interest in chemistry. her dedication to ensuring every student grasps the material is evident in the clarity of her explanations.
Super Tutor
Varthur, Bangalore
15 yrs of
Exp
9 students
497 hrs of classes completed
1000
per hour
Online Classes
I have over ten years of experience in the field of accounts. I am an IELTS-certified professional and have completed advanced Spoken and General English courses from the British Council. I hold a master's degree in Commerce and am interested in teaching. I have worked for Accenture, Deloitte, AIG, and a government-owned company, National Insurance. These experiences have given me real-time experience and expertise in the field of accounts, which I am eager to share with my students. As an IELTS 9 band scorer, I teach accounts and English to students in grades eleventh and twelfth. I offer Spoken English classes only online, one-on-one and in batches of up to tenth students. Each class lasts two hours a day, the introductory course lasts one and a half months, while the advanced course takes fifteen months to complete. For Spoken English, it takes around twenty to thirty hours to complete the basic level and forty-five to fifty hours for the advanced level. I accept students aged eleven years and above. I design a lesson plan that includes storytelling concepts, real-life examples, and practical exercises. I follow the Cambridge syllabus for accounts, and for English, I use practical-oriented techniques. I also frequently use a whiteboard, video, and audio links to help my students learn, and I provide soft copies of the materials, assignments, and question-and-answer sets for accounts.
His expertise shines through the clear explanations, making accounting concepts accessible and fostering a positive learning experience."
Top Tutor
Begur Road, Bangalore
5 yrs of
Exp
4 hrs of classes completed
350
per hour
Online Classes
Tutor's home
I have been teaching since last more than 5 years. My way of teaching style is so simple so that it make students very easier to understand. I am taking classes of Math and Hindi subject.
