Class 12 RS Aggarwal Chapter-30 Bayes’s Theorem and its Applications Solutions - Free PDF Download
FAQs on RS Aggarwal Class 12 Solutions Chapter-30 Bayes’s Theorem and its Applications
1. How should I approach solving problems from RS Aggarwal Class 12, Chapter 30, on Bayes's Theorem?
To solve problems from this chapter, you should follow a clear, step-by-step method. First, carefully read the problem to identify the set of mutually exclusive and exhaustive events (E₁, E₂, ..., Eₙ). Next, identify the event 'A' that has already occurred. Write down all the known probabilities, such as P(Eᵢ) and the conditional probabilities P(A|Eᵢ). Finally, state Bayes's Theorem formula and substitute the values to find the required posterior probability, P(Eᵢ|A).
2. What is the fundamental formula used to solve problems based on Bayes’s Theorem in Class 12 Maths?
The fundamental formula for Bayes's Theorem, which is essential for solving the exercises, is: P(Eᵢ|A) = [P(Eᵢ) * P(A|Eᵢ)] / Σ[P(Eₖ) * P(A|Eₖ)]. Here, P(Eᵢ|A) is the probability of an initial event Eᵢ given that event A has occurred. The denominator represents the theorem of total probability for event A. Correctly applying this formula is the key to every solution.
3. Why is it crucial to correctly define the initial events and the observed event before applying the formula?
Correctly defining the events is the most critical step. The initial events (E₁, E₂, ..., Eₙ) must form a partition of the sample space—meaning they are mutually exclusive and their union covers all possibilities. The observed event 'A' is the outcome you are given. If these are identified incorrectly, the entire structure of the solution, including the values of P(Eᵢ) and P(A|Eᵢ), will be wrong, leading to an incorrect final answer.
4. How do you solve a typical RS Aggarwal problem involving a doctor diagnosing a disease using Bayes's Theorem?
These problems are a classic application of the theorem. Follow these steps:
- Step 1: Define E₁ as the event that the person has the disease and E₂ as the event that the person does not. Note down their probabilities, P(E₁) and P(E₂).
- Step 2: Define 'A' as the event that the test result is positive.
- Step 3: Identify the conditional probabilities: P(A|E₁) (test is positive given the person has the disease - true positive) and P(A|E₂) (test is positive given the person does not have the disease - false positive).
- Step 4: Apply Bayes's Theorem to find the probability that the person actually has the disease given a positive test, i.e., P(E₁|A).
5. What is the main difference between a problem requiring the multiplication theorem of probability and one needing Bayes's Theorem?
The key difference lies in the question being asked. Use the multiplication theorem when you need to find the probability of a sequence of events occurring, like P(A and B). Use Bayes's Theorem when an event has already happened, and you need to find the probability of one of its specific causes. In essence, Bayes's Theorem calculates a 'reverse' conditional probability; you are working backward from an outcome to find the probability of a specific initial condition.
6. In the Bayes's Theorem formula, what is the role of the denominator, and why is it calculated using total probability?
The denominator, P(A), represents the total probability of the observed event 'A' occurring, regardless of the initial condition. It is calculated by summing the probabilities of 'A' occurring via each possible path: P(A) = P(E₁)P(A|E₁) + P(E₂)P(A|E₂) + ... + P(Eₙ)P(A|Eₙ). This sum acts as a normalizing factor, ensuring that the final calculated posterior probability P(Eᵢ|A) is a value between 0 and 1, making it a valid probability.
7. How does showing a step-by-step solution for Bayes's Theorem problems help in the CBSE Class 12 board exams?
As per the CBSE 2025-26 evaluation guidelines, marks in probability are awarded for specific steps. A clear, step-by-step solution ensures you get credit for:
- Correctly identifying all events and their probabilities.
- Stating the correct formula for Bayes's Theorem.
- Substituting the values accurately.
- Performing the final calculation correctly.
8. What is a common mistake students make when solving questions from RS Aggarwal Chapter 30?
A very common mistake is confusing P(A|B) with P(B|A). For example, in a disease-testing problem, students might mix up the probability of a test being positive given a person has the disease, with the probability of a person having the disease given the test is positive. Bayes's Theorem is specifically designed to calculate the latter based on the former. Always read the question carefully to distinguish between the prior probability and the posterior probability you need to find.
