Find the best tutors and institutes for Class 12 Tuition
Q3(i):
If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (i) $P(A \cap B)$
Solution :
Step 1: Given Variables and Theoretical Foundation
To determine the probability of the intersection of two events $A$ and $B$, we utilize the definition of conditional probability. The given parameters are:
- $P(A) = 0.8$
- $P(B) = 0.5$
- $P(B|A) = 0.4$
The conditional probability of event $B$ given that event $A$ has occurred is defined as: [By the Definition of Conditional Probability]
$P(B|A) = \frac{P(A \cap B)}{P(A)}$, where $P(A) > 0$.
Step 2: Rearrangement of the Formula
To isolate the target variable $P(A \cap B)$, we multiply both sides of the equation by $P(A)$:
$P(A \cap B) = P(B|A) \times P(A)$
Step 3: Numerical Substitution and Calculation
Substitute the provided values into the derived formula:
$P(A \cap B) = 0.4 \times 0.8$
$P(A \cap B) = 0.32$
Visual Representation of Sample Space
The intersection $P(A \cap B)$ represents the probability that both events $A$ and $B$ occur simultaneously. Based on the calculation derived from the conditional probability definition, the product of the probability of $A$ and the conditional probability of $B$ given $A$ yields the joint probability.
Final Solution: $P(A \cap B) = 0.32$
More Questions from Class 12 Mathematics Probability EXERCISE 13.1
- Q1: Given that E and F are events such that $P(E) = 0.6$, $P(F) = 0.3$ and $P(E \cap F) = 0.2$, find $P(E|F)$ and $P(F|E)$
- Q2: Compute $P(A|B)$, if $P(B) = 0.5$ and $P(A \cap B) = 0.32$
- Q3(ii): If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (ii) $P(A|B)$
- Q3(iii): If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (iii) $P(A \cup B)$
- Q4: Evaluate $P(A \cup B)$, if $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$
- Q5(i): If $P(A) = \frac{6}{11}$ , $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$, find (i) $P(A\cap B)$
- Q5(ii): If $P(A) = \frac{6}{11}$ , $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$, find (ii) $P(A|B)$
- Q5(iii): If $P(A) = \frac{6}{11}$ , $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$, find (iii) $P(B|A)$
- Q6(i): Determine $P(E|F)$. A coin is tossed three times, where (i) E : head on third toss, F : heads on first two tosses
- Q6(ii): Determine $P(E|F)$. A coin is tossed three times, where (ii) E : at least two heads, F : at most two heads
- Q6(iii): Determine $P(E|F)$. A coin is tossed three times, where (iii) E : at most two tails, F : at least one tail
- Q7(i): Determine $P(E|F)$. Two coins are tossed once, where (i) E : tail appears on one coin, F : one coin shows head
- Q7(ii): Determine $P(E|F)$. Two coins are tossed once, where (ii) E : no tail appears, F : no head appears
- Q8: Determine $P(E|F)$. A die is thrown three times, E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses
- Q9: Determine $P(E|F)$. Mother, father and son line up at random for a family picture E : son on one end, F : father in middle
- Q10(a): A black and a red dice are rolled. (a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
- Q10(b): A black and a red dice are rolled. (b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
- Q11(i): A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$. Find (i) $P(E|F)$ and $P(F|E)$
- Q11(ii): A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$. Find (ii) $P(E|G)$ and $P(G|E)$
- Q11(iii): A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$. Find (iii) $P((E \cup F)|G)$ and $P((E \cap F)|G)$
- Q12(i): Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl,
- Q12(ii): Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (ii) at least one is a girl?
- Q13: An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?
- Q14: Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.
- Q15: Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.
- Q16: If $P(A) = \frac{1}{2}$, $P(B) = 0$, then $P(A|B)$ is
- Q17: If A and B are events such that $P(A|B) = P(B|A)$, then
CBSE Solutions for Class 12 Mathematics Probability
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Probability
Super Tutor
Shahdara Chitrakoot, Delhi
8 yrs of
Exp
14 students
7171 hrs of classes completed
500
per hour
Online Classes
With years of experience in providing Class 12 tuition, I specialize in teaching both mathematics and computer science. My approach is tailored to meet the rigorous demands of the Class 12 curriculum, focusing on in-depth understanding and application of concepts. In mathematics, I cover a broad spectrum of topics including calculus, algebra, trigonometry, and statistics, ensuring students grasp the fundamental theories and their practical applications. In computer science, my teaching encompasses programming languages such as Java and C++, data structures, algorithms, and database management. I integrate real-world problem-solving techniques and coding practices to prepare students for both exams and future academic endeavors. My lessons are structured to build a strong foundation in both theoretical and practical aspects, using a blend of traditional teaching methods and modern technologies. I employ interactive tools like educational software, online coding platforms, and simulation tools to enhance learning. My classes are supplemented with detailed notes, well-structured PowerPoint presentations, and regular assessments to track progress. I also provide personalized attention to address individual student needs and learning paces. Regular doubt-clearing sessions, practice tests, and revision modules are key components of my teaching strategy. I ensure students are well-prepared for their board exams and competitive entrance tests, emphasizing conceptual clarity and efficient problem-solving techniques. My commitment is to foster a deep understanding and passion for mathematics and computer science, equipping students with the skills and knowledge to excel academically and beyond.
I’ve had an excellent learning experience with Rahul Sir. One of the commendable aspects is that he completed the syllabus well before time, which will provide me with ample opportunity to review and reinforce my understanding of the concepts. He articulates topics with clarity, ensures that the fundamentals are well understood, and always takes time to clear doubts patiently. His teaching style is systematic and engaging, which makes even difficult topics easier to grasp. His mentorship has improved my speed and accuracy. Additionally, I appreciate his punctuality, structured planning of lessons, and the way he motivates students to stay consistent. Overall, he is a dedicated teacher who truly cares about his students’ progress, and I would highly recommend him to anyone seeking quality Math tuition.
Alamganj, Patna
10 yrs of
Exp
Online Classes
I am an experienced English language and literature faculty. I completed my masters in MA from Patna University in the year 2016 and currently pursuing a PhD in English Literature. I am working as a Speaking Examiner with Cambridge Assessment English, UK. I have 10 years of experience in teaching Spoken English, BCom, BCA, BSc, BTech and Class X-XII selected subjects for ISC/ICSE, State and CBSE Board students. I have taught students of different age groups and demography. My classes are interactive and learner-oriented. I follow various aspects of training with special Emphasis on Pronunciation.
Top Tutor
Jahangirpura, Surat
15 yrs of
Exp
6 students
3687 hrs of classes completed
250
per hour
Online Classes
I have 15 years of teaching Maths to students of clasd 11,12 Science. I have taught NCERT and RD Sharma.
The classes are going well. I am getting a better clarity on the concepts which were already taught. Sir is giving enough time for me to understand better.
Top Tutor
Sector 55 Phase 1, Sahibzada Ajit Singh Nagar
16 yrs of
Exp
2 students
62 hrs of classes completed
500
per hour
Online Classes
Tutor's home
I graduated in Chemical engineering. I have eighteen years of teaching experience. I have taught in well-known and leading institutes. I am passionate about maths and love the subject. My passion for the subject motivated me to teach others. I teach math for classes ninth to twelfth and Engineering entrance. My classes are conducted online and offline. My classes are practical, where I simplify students' various challenges. I also interact with students and understand their views. My methods are simple and easy for the students to grasp. I also extensively use whiteboards and Powerpoint presentations for explanations. I also offer self-made assignments for students to practice. Periodic tests are conducted for the students to track their improvements and growth.
Top Tutor
Domlur, Bangalore
15 yrs of
Exp
31 students
3092 hrs of classes completed
700
per hour
Online Classes
Taught college students for more than 15 years. I have taught several subjects. I also teach for competitive exams like NEET, PET, BITSAT, GATE ,CET. I also teach a variety of students starting from Class 10 to M.Tech for the engineering stream. I also prepare students for several competitive exams. I also guide them how to crack any exam.
Very dedicated and sincere about teaching the subject, and creates a very comfortable atmosphere in all his classes. he teaches with utmost patience and clears all doubts without judgement or hesitation. He explains concepts without rushing and gives practice questions in class, which are very helpful.
