Find the best tutors and institutes for Class 12 Tuition
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?
Solution :
Step 1: Defining Events and Given Probabilities
Let the sample space be the set of all students in the college. We define the following events:
- $H$: The event that the chosen student resides in a hostel.
- $D$: The event that the chosen student is a day scholar.
- $A$: The event that the chosen student attains an A grade.
Based on the provided data, we assign the following probabilities:
- $P(H) = 0.60$ [Given: 60% of students are hostellers]
- $P(D) = 0.40$ [Given: 40% of students are day scholars]
- $P(A|H) = 0.30$ [Given: 30% of hostellers attain A grade]
- $P(A|D) = 0.20$ [Given: 20% of day scholars attain A grade]
Step 2: Visualization of Probability Space
Step 3: Calculating Total Probability of Event A
To find the probability that a randomly chosen student has an A grade, we use the Law of Total Probability [By Theorem of Total Probability]:
$P(A) = P(H) \cdot P(A|H) + P(D) \cdot P(A|D)$
$P(A) = (0.60 \times 0.30) + (0.40 \times 0.20)$
$P(A) = 0.18 + 0.08$
$P(A) = 0.26$
Step 4: Applying Bayes' Theorem
We are required to find the probability that the student is a hosteller, given that they have an A grade. This is represented by $P(H|A)$. [By Bayes' Theorem]:
$P(H|A) = \frac{P(H) \cdot P(A|H)}{P(A)}$
$P(H|A) = \frac{0.60 \times 0.30}{0.26}$
$P(H|A) = \frac{0.18}{0.26}$
Step 5: Simplification of the Fraction
$P(H|A) = \frac{18}{26} = \frac{9}{13}$
Converting to a decimal (approximate): $P(H|A) \approx 0.6923$
Final Solution: The probability that the student is a hosteller is $\frac{9}{13}$.
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.
- 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
- 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?
CBSE Solutions for Class 12 Mathematics Probability
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Probability
Top Tutor
Noida Industrial Area, Noida
10 yrs of
Exp
5 students
98 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I have both experience and knowledge in tutoring of Mathematics. I have been giving tutoring for last 10 years. I have a great academic record both at school and college level. I believe in making the student understand the concept which is important from exam point of view also and also for future growth of student Personal attention with quality learning can provide you better understanding of concepts. This is what I want to ensure to the students so that I will be able to make difference between their past, present and future. I am also a good motivator and a creative person as well. So, it will help the students to learn the concepts clearly. I find satisfaction in making students understand their subjects by making them finding their own areas of strength. As I qualified GATE and have a good exposure of lab so I know the level of topics and questions ask in competitive exams. I am motivated person so always try to appreciate and boost my students. Good in communications. My focus will be on developing a strong base of each and every subject taught to the student. Innovative techniques of learning will be introduced based on the efficiency of the student to grasp the subject matter. Another area of focus would be helping the students to strategies methodology for self-study purpose. Adequate time will be provided and clearing doubts will always be the top priority. Fluency in all subjects is assured.
Top Tutor
R - Block, Kovaipudur
20 yrs of
Exp
16 students
296 hrs of classes completed
500
per hour
Online Classes
I have 21 years of experience in teaching chemistry and training the students for both board and competitive exams.
Fantastic teacher with great experience in her field. Definitely impressed. Best for organic chemistry.
Super Tutor
B-seam Colly, Manendragarh
2 yrs of
Exp
3 students
132 hrs of classes completed
350
per hour
Online Classes
Tutor's home
I am a chemistry teacher. I am given home tuition since 2022. I am certified in 2022. I have a degree in MSC chemistry. My key skills is making complicated topic chemistry topics easy to understanding for students.
He is a focused tracher . He has a good conceptual knowledge. He teaches according to student expectations. Very patient with student. He is regular also in giving classes.
Top Tutor
KR Puram, Bangalore
8 yrs of
Exp
2 students
31 hrs of classes completed
500
per hour
Online Classes
I am a physics teacher handling X,XI,XII grades, having 8 years experience as Science HOD in a cbse School. I have a degree in BE ( electronics and telecommunications).With my best efforts, support, patience ,commitment I unlock my students potential.Concepts are presented in a clear, simple and direct manner. Consistently my class result is 100% with good subject average. Expanding pupils horizons is my accomplishment.
The teaching is really good and very understandable. Mam makes concept learning very easy even for the slow learners. Very happy to be a part of it.
Top Tutor
Mukherjee Nagar, Delhi
5 yrs of
Exp
1 student
2040 hrs of classes completed
500
per hour
Online Classes
Hi I am Neeraj Dwivedi Top Tutor and partner at urban pro.Into profession of teaching since last four years. Expertise and specialization in guiding students on subjects like History, Political Science , Economics and Hindi. These subjects were my strength in my journey of Civil Services and later helped in shaping future of students. Apart from teaching guidance I have been doing career counselling of students as to what to pursue/not pursue after school based on individual interests and capabilities.
