RD Sharma Class 12 Solutions Chapter 31 - Probability (Ex 31.4) Exercise 31.4 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 31 - Probability (Ex 31.4) Exercise 31.4
1. What is the fundamental step-by-step method for solving questions in RD Sharma Class 12 Solutions Ex 31.4 that involve the Theorem of Total Probability?
To correctly solve problems using the Theorem of Total Probability in this exercise, follow these steps as per the CBSE curriculum:
- First, identify a set of mutually exclusive and exhaustive events (e.g., E₁, E₂, E₃) that form a partition of the sample space. For example, selecting Urn 1, Urn 2, or Urn 3.
- Determine the probability of each of these initial events: P(E₁), P(E₂), P(E₃).
- Identify the final event 'A' whose probability you need to find (e.g., drawing a red ball).
- Calculate the conditional probability of 'A' for each initial event: P(A|E₁), P(A|E₂), P(A|E₃).
- Finally, apply the Theorem of Total Probability formula: P(A) = P(E₁)P(A|E₁) + P(E₂)P(A|E₂) + P(E₃)P(A|E₃).
2. How should Bayes' Theorem be applied to find a reverse probability in problems from Exercise 31.4?
Bayes' Theorem is used when the outcome of an experiment is known, and you need to find the probability of the initial event that led to it. The correct method is:
- Define the initial events (hypotheses) E₁, E₂, ..., Eₙ and the observed final event A.
- List the probabilities of the initial events, P(Eᵢ).
- List the conditional probabilities of the observed event A given each hypothesis, P(A|Eᵢ).
- The goal is to find P(Eᵢ|A), i.e., the probability of a specific initial event given A occurred.
- Apply the Bayes' Theorem formula: P(Eᵢ|A) = [P(Eᵢ) × P(A|Eᵢ)] / Σ[P(Eₖ) × P(A|Eₖ)]. The denominator is simply the Total Probability of A.
3. What is a common point of confusion when solving questions in RD Sharma Exercise 31.4, and how can it be avoided?
A common confusion for students is mixing up when to use the Theorem of Total Probability versus Bayes' Theorem. To avoid this, analyse what the question asks for:
- If it asks for the overall probability of a final outcome (e.g., "Find the probability that the ball drawn is black"), use the Theorem of Total Probability.
- If it provides the final outcome as a fact (e.g., "A black ball was drawn") and asks for the probability of a specific initial cause (e.g., "...what is the probability it was drawn from the first bag?"), you must use Bayes' Theorem.
4. Why is calculating the 'total probability' often a prerequisite for applying Bayes' Theorem in this exercise?
In the formula for Bayes' Theorem, P(Eᵢ|A) = [P(Eᵢ) × P(A|Eᵢ)] / P(A), the denominator P(A) represents the total probability of event A occurring. This P(A) is calculated using the Theorem of Total Probability (Σ[P(Eₖ) × P(A|Eₖ)]). Therefore, solving for the total probability of the observed event is a necessary intermediate step before you can determine the final reverse probability required by Bayes' Theorem.
5. How do you correctly interpret the different probability terms in a typical Bayes' Theorem problem from Exercise 31.4, such as a factory manufacturing bolts?
In a typical problem where machines M₁, M₂, and M₃ produce bolts, the terms are interpreted as follows:
- P(M₁): This is the prior probability. It represents the base chance of selecting a bolt made by Machine M₁ before you know if it's defective. For instance, if M₁ makes 30% of all bolts, P(M₁) = 0.3.
- P(Defective|M₁): This is the conditional probability. It is the probability that a bolt is defective, given that it was made by Machine M₁.
- P(M₁|Defective): This is the posterior probability calculated using Bayes' Theorem. It represents the updated probability that a bolt came from Machine M₁ after you have observed that it is defective.
6. For a question in Exercise 31.4 about a medical diagnosis, what is the conceptual difference between the probability of a positive test for a patient with the disease versus the probability that a patient has the disease given a positive test?
These two probabilities address different questions and are often confused:
- P(Positive Test | Disease): This is the test's accuracy or sensitivity. It answers: "If a person has the disease, what is the chance the test will correctly identify it?" This value is usually given in the problem.
- P(Disease | Positive Test): This is the diagnostic value of the test, calculated using Bayes' Theorem. It answers: "If a person's test comes back positive, what is the actual chance they have the disease?" This value depends not only on test accuracy but also on the overall prevalence of the disease in the population.
7. How does the structure of problems in RD Sharma Ex 31.4 relate to the CBSE Class 12 syllabus for the 2025-26 session?
The problems in RD Sharma Exercise 31.4 are designed to provide extensive practice on core concepts from the CBSE Class 12 Maths syllabus for 2025-26. This exercise specifically focuses on the application of the Theorem of Total Probability and Bayes' Theorem, which are high-importance topics within the Probability unit. Mastering the step-by-step solutions for this exercise ensures a strong foundation for solving complex conditional probability questions that frequently appear in CBSE board exams.
