Find the best tutors and institutes for Class 12 Tuition
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?
Solution :
Step 1: Definition of Events
Let the sample space be the set of all items produced by the factory. We define the following events:
- $E_1$: The event that the item is produced by Machine A.
- $E_2$: The event that the item is produced by Machine B.
- $A$: The event that the chosen item is defective.
Step 2: Assigning Probabilities to Events
Based on the provided data, we define the prior probabilities [Per the Law of Total Probability]:
- $P(E_1) = 60\% = 0.60$
- $P(E_2) = 40\% = 0.40$
The conditional probabilities of producing a defective item given the machine are:
- $P(A|E_1) = 2\% = 0.02$
- $P(A|E_2) = 1\% = 0.01$
Step 3: Calculating Total Probability of a Defective Item
The total probability $P(A)$ that an item is defective is given by the formula [By the Law of Total Probability]:
$P(A) = P(E_1) \cdot P(A|E_1) + P(E_2) \cdot P(A|E_2)$
Substituting the values:
$P(A) = (0.60 \times 0.02) + (0.40 \times 0.01)$
$P(A) = 0.012 + 0.004 = 0.016$
Step 4: Applying Bayes' Theorem
We are required to find the probability that the defective item came from Machine B, which is $P(E_2|A)$. [By Bayes' Theorem]:
$P(E_2|A) = \frac{P(E_2) \cdot P(A|E_2)}{P(E_1) \cdot P(A|E_1) + P(E_2) \cdot P(A|E_2)}$
$P(E_2|A) = \frac{0.40 \times 0.01}{0.016}$
$P(E_2|A) = \frac{0.004}{0.016}$
$P(E_2|A) = \frac{4}{16} = \frac{1}{4} = 0.25$
Visual Representation of Probability Distribution
Final Solution: The probability that the defective item was produced by machine B is $0.25$ or $25\%$.
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?
- 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
Kamal Chowk, Nagpur
20 yrs of
Exp
Tutor's home
I have 20 years of experience in the field of teaching. I conduct classes for Spoken English, BBA, MBA, Hindi Language, Career Counselling and Classes I to X. I tutor (offline/online) for syllabi-based subjects Classes I to X for CBSE, ICSE, IGCSE, IB Board. For classes X and above, I focus on tutoring subjects like Hindi, English, Psychology. Accountancy, Business Studies, Mathematics, Physics, Chemistry for CBSE, ICSE, IGCSE and IB Board. I also conduct career counselling and psychological counselling. I also assist students in pursuing MBA/PGDBM courses in their dissertation work. People engaged in doctoral theses in the domains of management and psychology engage my services for their research work. Since there is a lot of demand for enhancing one's conversational English, this is what I am focusing on at the moment. Being bilingual, my medium of communication is English as well as Hindi. I train learners in the usage of WordPress, Joomla - to develop their websites; right from domain registration to site maintenance. Besides, I assist teachers in pursuing Teachers' Training Courses.
Top Tutor
Uttam Nagar, Delhi
10 yrs of
Exp
2 students
132 hrs of classes completed
1000
per hour
Online Classes
Student's Home
Efficient professional with 9+ years of experience and proven knowledge of lectures, training and development, and adaptive teaching methods. I aim to leverage my skills to successfully fill the significant role of a teacher in your life. Also, through my experience, I have developed interactive teaching methods that can help you understand every concept's very applicability, enabling you to correlate the readings with practical situations.
I will always be grateful for the time and effort you put into teaching me. Your passion for education is truly infectious!
Top Tutor
Electronic City, Bangalore
11 yrs of
Exp
5 students
185 hrs of classes completed
500
per hour
Online Classes
Well experienced in physics and chemistry of XI , XII standard in both ICSE and CBSE curricular. Taken part time classes in college for ISC syllabus. Strong hold on Mechanics and Organic compounds and thermodynamics.
Top Tutor
Bhopal, Bhopal
4 yrs of
Exp
1 student
75 hrs of classes completed
300
per hour
Online Classes
I am a mechanical engineer currently I am in pre final year in NIT BHOPAL. I have cracked jee on my own in first attempt and I also scored 90% in 12 with cbse board. I have been teaching my friends from class 11th itself. I can make subject really easy for you.
Top Tutor
Kota, Kota
1 yrs of
Exp
3 students
61 hrs of classes completed
500
per hour
Online Classes
I have one year of teaching experience at the university level, where I taught PG (M.Com) students in both offline and online modes. I am UGC NET qualified in Commerce, which has strengthened my subject knowledge and conceptual understanding. Teaching at the postgraduate level has helped me develop clarity in core commerce subjects. I am confident that I can explain concepts in a simple and effective manner and adapt my teaching style to suit the learning needs of Class 12 students as per the CBSE syllabus.
Ma’am taught me Accountancy for class 12. Her teaching style is easy to understand and exam-oriented. She provides good notes and practice questions, which helped me improve my performance.”
