Find the best tutors and institutes for Class 12 Tuition
Q14:
If A and B are two events such that $A \subset B$ and $P(B) \neq 0$, then which of the following is correct?
Solution :
Step 1: Theoretical Analysis of the Subset Condition
Given the condition $A \subset B$, by the definition of a subset in probability theory, every sample point in event $A$ is also contained within event $B$. Consequently, the intersection of $A$ and $B$ is simply $A$:
$A \cap B = A$ [Definition of Subset]
Therefore, the probability of the intersection is:
$P(A \cap B) = P(A)$
Step 2: Evaluating Conditional Probability
The conditional probability of $A$ given $B$ is defined by the formula [Per the Axioms of Probability]:
$P(A|B) = \frac{P(A \cap B)}{P(B)}$
Substituting the result from Step 1 into this formula:
$P(A|B) = \frac{P(A)}{P(B)}$
Step 3: Logical Deduction of the Inequality
Since $A \subset B$, it follows that $P(A) \leq P(B)$ [Monotonicity Property of Probability]. Given $P(B) \neq 0$, we analyze the possible values for $P(A|B)$:
- Since $P(A) \leq P(B)$, then $\frac{P(A)}{P(B)} \leq 1$.
- Since $P(A) \geq 0$ and $P(B) > 0$, then $\frac{P(A)}{P(B)} \geq 0$.
Step 4: Comparative Analysis of Options
In standard textbook problems of this type, the objective is to identify the relationship between $P(A|B)$ and $P(A)$.
| Expression | Logical Status |
|---|---|
| $P(A|B) \geq P(A)$ | Correct (Since $P(B) \leq 1$, dividing $P(A)$ by a value $\leq 1$ increases or maintains the value) |
| $P(A|B) < P(A)$ | Incorrect |
| $P(A|B) = P(B)/P(A)$ | Incorrect |
Final Solution: The correct relationship is $P(A|B) \geq P(A)$, as $P(A|B) = \frac{P(A)}{P(B)}$ and $P(B) \leq 1$.
More Questions from Class 12 Mathematics Probability EXERCISE 13.3
- Q1: An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
- Q2: A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
- Q3: Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?
- Q4: In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\frac{3}{4}$ be the probability that he knows the answer and $\frac{1}{4}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$. What is the probability that the student knows the answer given that he answered it correctly?
- Q5: A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive ?
- Q6: There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?
- Q7: An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
- Q8: A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
- Q9: Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
- Q10: Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
- Q11: A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
- Q12: A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
- Q13: Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is
CBSE Solutions for Class 12 Mathematics Probability
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Probability
Top Tutor
Sector 33, Gurgaon
18 yrs of
Exp
4 students
763 hrs of classes completed
1200
per hour
Online Classes
Tutor's home
Expert Chemistry Mentor for IIT-JEE | NEET | CBSE – 18+ Years of Experience Hi, I’m Anuranjita. I am a Senior Chemistry Expert with over 18 years of experience at India's leading institutes like Aakash, FIITJEE, and TIME and hold an M.Tech. degree from Anna University, Chennai—one of the top 10 engineering universities in India. I have refined a highly structured, result-oriented teaching methodology designed to crack • IIT-JEE (Main & Advanced) • NEET • Score 100% in CBSE Board exams. (Class 11 & 12 – Chemistry) Over the years, I have trained thousands of students, many of whom have successfully secured admissions into AIIMS, IITs, NITs, and other top medical and engineering colleges across India. Here is what makes my classes uniquely effective: • Interactive and concept-driven teaching • Focus on building strong fundamentals • Personalized learning based on each student’s pace • Regular doubt-clearing sessions & Chapter-wise tests • Exam-oriented tips and problem-solving strategies • The "Exclusive 5" Batch Model (Personalized Attention) Unlike massive online batches with thousands of students where individual voices get lost, my classes are strictly limited to a maximum of 5 students. This ensures that every single student receives 1-on-1 level attention, and I can monitor each student's real-time progress, strengths, and weaknesses. • Rigorous Testing & PYQ Practice: Chapter-wise tests modeled strictly on the latest NTA patterns, alongside exhaustive solving of past years' questions (PYQs). Looking forward to helping you master chemistry! Best Regards Anuranjita
The class was good. The explanation of the concept is clear. All the queries are being addressed to the student accordingly.
Top Tutor
Mahipalpur, Delhi
14 yrs of
Exp
2 hrs of classes completed
Student's Home
As an experienced teacher for 12 years, I have been teaching to make students interested in the subjects of Maths. My attitude is to help children improve in studies in the best possible way.
Top Tutor
P847+PC6 Thrissur, Thrissur
17 yrs of
Exp
10 students
501 hrs of classes completed
600
per hour
Online Classes
I believe in conceptual learning and always relate the content to real life situations. I discourage the rote learning and promote the application of the concept in daily life situation.
I have enjoyed madam's classes a lot. Her sessions are very engaging. She has helped me to understand even the difficult concepts easily.
Top Tutor
Biman Nagar, Kolkata
8 yrs of
Exp
500
per hour
Online Classes
Student's Home
Tutor's home
I am actively teaching physics up to class XII ( including JEE and NEET aspirants) from last 8 years. I run my own Tuition center where I provide physics class to NEET and IIT JEE aspirants and provide home Tuitions to deserving students. There is a separate doubt clearance & numerical problem solving batch for candidate who already joined Aakash, fiit jee, Allen and other coaching institutes.
Everything that sir taught me was easy to grasp and learning was fun. It helped me a lot during my exams. I became immaculate at all the topics taught to me by him because of the flawless way of his teaching. I can never thank Sir enough for what he did for me. He was also a mentor and inspiration for me.
Top Tutor
Pandav Nagar, Ghaziabad
9 yrs of
Exp
10 students
22 hrs of classes completed
600
per hour
Tutor's home
I am an experienced, qualified teacher with over 9 years of experience in teaching Math across different boards including CBSE, ICSE and IGCSE and state board 9-12 and IIT JEE engineering entrance mathematics . I have worked in reputed institutes like Vibrant academy, Resonance in Kota and in Byjus and Modi School in Rajkot. Passionate about solving mathematical problems. And I have also been carrying out year of experience in Engineering Mathematics.
Eti mam is a good teacher. She have a different way of teaching. I am glad to improved grades of my son.
