Probability Class 12th Ncert Solutions

Probability Class 12th NCERT Solutions: A Comprehensive Guide

Understanding probability can seem daunting, but with a systematic approach and the right resources, mastering this crucial topic in Class 12th mathematics becomes achievable. This article provides comprehensive solutions and explanations for the NCERT (National Council of Educational Research and Training) textbook problems on probability, aiming to build a strong foundation for your understanding. We'll cover key concepts, solve example problems, and address common student queries, ensuring you're well-prepared for your examinations. This guide focuses on building conceptual clarity alongside problem-solving skills, making probability less intimidating and more engaging.

Introduction to Probability

Probability, at its core, deals with the likelihood of an event occurring. It quantifies uncertainty, assigning numerical values between 0 and 1 to represent the chance of an event happening. 0 indicates impossibility, while 1 represents certainty. The NCERT Class 12th textbook covers various aspects of probability, including:

• Sample Space: The set of all possible outcomes of a random experiment.
• Events: Subsets of the sample space.
• Probability of an Event: The ratio of the number of favorable outcomes to the total number of possible outcomes.
• Conditional Probability: The probability of an event occurring given that another event has already occurred.
• Independent Events: Events whose occurrence does not affect the probability of the other event occurring.
• Bayes' Theorem: A formula for calculating conditional probabilities.
• Random Variables: Variables whose values are numerical outcomes of a random phenomenon.
• Probability Distributions: Functions that assign probabilities to different values of a random variable. This includes the binomial distribution which is frequently covered in the NCERT textbook.

Key Concepts and Formulae

Before diving into specific problems, let's review some essential formulas and concepts:

• Probability of an event A: P(A) = (Number of favorable outcomes for A) / (Total number of possible outcomes)
• Probability of the complement of A (A'): P(A') = 1 - P(A)
• Addition Theorem of Probability: P(A∪B) = P(A) + P(B) - P(A∩B) (For any two events A and B)
• Conditional Probability of A given B: P(A|B) = P(A∩B) / P(B), provided P(B) > 0
• Multiplication Theorem of Probability: P(A∩B) = P(A)P(B|A) = P(B)P(A|B)
• Independent Events: If A and B are independent, then P(A∩B) = P(A)P(B)

Solving NCERT Problems: A Step-by-Step Approach

The NCERT textbook presents a range of problems, from straightforward calculations to more complex scenarios involving conditional probability and Bayes' theorem. Let's address some example problems, illustrating the step-by-step approach to solving them:

Example 1: Simple Probability

Problem: A bag contains 5 red balls and 3 blue balls. What is the probability of drawing a red ball?

Solution:

1. Identify the favorable outcomes: Drawing a red ball. There are 5 red balls.
2. Identify the total number of possible outcomes: Total number of balls is 5 + 3 = 8.
3. Apply the probability formula: P(Red) = (Number of red balls) / (Total number of balls) = 5/8

Example 2: Conditional Probability

Problem: A box contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that the second marble is red given that the first marble is blue?

Solution:

1. Define events: Let A be the event that the first marble is blue, and B be the event that the second marble is red.
2. Calculate P(A): P(A) = 6/10 = 3/5 (6 blue marbles out of 10 total)
3. Calculate P(B|A): If the first marble is blue, there are 9 marbles left, 4 of which are red. Therefore, P(B|A) = 4/9
4. Calculate P(B∩A): Using the multiplication theorem: P(B∩A) = P(A)P(B|A) = (3/5) * (4/9) = 4/15

Therefore, the probability that the second marble is red given that the first marble is blue is 4/9.

Example 3: Bayes' Theorem

Problem: A diagnostic test for a disease has a 90% accuracy rate for detecting the disease when it is present and a 95% accuracy rate for correctly identifying those who don't have the disease. If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease?

Solution:

This problem requires Bayes' Theorem. Let's define the events:

• D: Person has the disease
• D': Person does not have the disease
• T: Test is positive
• T': Test is negative

We are given:

• P(D) = 0.01 (1% prevalence)
• P(D') = 0.99
• P(T|D) = 0.90 (90% sensitivity)
• P(T'|D') = 0.95 (95% specificity) This means P(T|D') = 1 - 0.95 = 0.05

We want to find P(D|T). Bayes' Theorem states:

P(D|T) = [P(T|D) * P(D)] / [P(T|D) * P(D) + P(T|D') * P(D')]

Plugging in the values:

P(D|T) = (0.90 * 0.01) / (0.90 * 0.01 + 0.05 * 0.99) ≈ 0.1538

Therefore, even with a seemingly accurate test, only about 15.38% of those testing positive actually have the disease. This highlights the importance of understanding the context and limitations of probability in real-world applications.

Common Mistakes to Avoid

Students often make these mistakes when solving probability problems:

• Confusing independent and dependent events: Remember to check if events affect each other before applying the appropriate formula.
• Incorrectly applying the addition theorem: Remember to subtract the probability of the intersection if events are not mutually exclusive.
• Misinterpreting conditional probability: Carefully identify the given condition and apply the correct formula.
• Not accounting for replacement: If items are drawn without replacement, the probabilities change for subsequent draws.

Frequently Asked Questions (FAQ)

Q1: What is the difference between permutations and combinations in probability?

Permutations consider the order of arrangements, while combinations do not. For example, arranging 3 books on a shelf is a permutation problem, while selecting 3 books from a set of 10 is a combination problem.

Q2: How can I improve my problem-solving skills in probability?

Practice is key! Work through as many problems as possible from the NCERT textbook and other resources. Focus on understanding the underlying concepts and systematically applying the appropriate formulas. Visual aids like Venn diagrams can be helpful for understanding set theory concepts related to probability.

Q3: What resources are available besides the NCERT textbook?

Numerous online resources, practice books, and video tutorials can supplement your learning. Look for resources that explain concepts clearly and provide step-by-step solutions to problems.

Conclusion

Mastering probability requires a solid understanding of fundamental concepts, systematic problem-solving skills, and consistent practice. The NCERT Class 12th textbook provides a strong foundation, and this article aims to guide you through the key concepts and problem-solving strategies. By understanding the nuances of probability, and applying the formulas correctly, you can confidently tackle any problem presented in your examinations. Remember that consistent practice and a clear grasp of the underlying principles are the cornerstones of success in this fascinating branch of mathematics. Don't hesitate to revisit this guide and refer to the NCERT textbook for further clarification whenever needed. Good luck!