NCERT Solutions for Class 12 Maths Chapter 13 – Probability: A Comprehensive Guide

Introduction

Mathematics is a crucial part of the academic curriculum, especially for students in Class 12 who are preparing for board exams and various competitive tests. Chapter 13 of the Class 12 NCERT Mathematics textbook, titled "Probability," is one of the key chapters that introduces students to fundamental concepts in probability theory. This chapter is designed to equip students with the skills to understand and solve problems related to probability, which is essential for their academic success.

What is Probability?

Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes. It quantifies the chance of an event occurring and is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. Understanding probability is crucial as it has real-world applications in fields such as statistics, finance, insurance, and various scientific disciplines.

Key Concepts in Chapter 13 – Probability

Chapter 13 of the Class 12 NCERT Mathematics textbook covers several important topics in probability, including:

1. Probability of an Event: Basic definition and concepts of probability.
2. Conditional Probability: Probability of an event given that another event has already occurred.
3. Multiplication Theorem: Rules for calculating the probability of the intersection of two events.
4. Bayes' Theorem: A theorem used to update the probability of an event based on new information.
5. Random Variables and Probability Distributions: Introduction to random variables and their probability distributions.
6. Binomial Distribution: A special type of probability distribution that describes the number of successes in a fixed number of independent trials.

1. Probability of an Event

The probability of an event $A$ is defined as the ratio of the number of favorable outcomes to the number of possible outcomes in a sample space. Mathematically, it is represented as:

$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

For example, when rolling a fair die, the probability of getting a 4 is:

$P(\text{Rolling a 4}) = \frac{1}{6}$

Here, 1 is the number of favorable outcomes (rolling a 4) and 6 is the total number of possible outcomes (sides of the die).

2. Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$, which represents the probability of event $A$ occurring given that event $B$ has already occurred. The formula for conditional probability is:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

where $P(A \cap B)$ is the probability of both events $A$ and $B$ occurring simultaneously, and $P(B)$ is the probability of event $B$ occurring.

For instance, if a deck of cards is shuffled and one card is drawn, the probability of drawing an Ace given that the card drawn is a spade is:

$P(\text{Ace} | \text{Spade}) = \frac{\text{Number of Aces in Spades}}{\text{Total number of Spades}} = \frac{1}{13}$

3. Multiplication Theorem

The multiplication theorem of probability provides a method for finding the probability of the intersection of two events. There are two cases:

1. For Independent Events: If events $A$ and $B$ are independent, then:

$P(A \cap B) = P(A) \times P(B)$

2. For Dependent Events: If events $A$ and $B$ are not independent, then:

$P(A \cap B) = P(A) \times P(B|A)$

For example, if you roll two dice, the probability of getting a 6 on both dice (independent events) is:

$P(\text{6 on first die} \cap \text{6 on second die}) = P(\text{6 on first die}) \times P(\text{6 on second die}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

4. Bayes' Theorem

Bayes' Theorem is a fundamental theorem in probability theory used to update the probability of an event based on new information. It is expressed as:

$P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$

where:

• $P(A|B)$ is the posterior probability (the probability of event $A$ given $B$).
• $P(B|A)$ is the likelihood (the probability of observing $B$ given $A$).
• $P(A)$ is the prior probability (the initial probability of $A$).
• $P(B)$ is the marginal likelihood (the total probability of $B$).

For instance, if you have a test that detects a certain disease with a known probability of accuracy, Bayes' Theorem can help you determine the probability of having the disease given a positive test result.

5. Random Variables and Probability Distributions

A random variable is a variable whose value is subject to variations due to chance. There are two types of random variables:

1. Discrete Random Variables: These can take on a countable number of values. For example, the number of heads in 10 coin flips is a discrete random variable.

2. Continuous Random Variables: These can take on an infinite number of values within a range. For example, the height of a person is a continuous random variable.

The probability distribution of a random variable provides a way to describe how probabilities are distributed over the values of the random variable. For discrete random variables, this is represented by a probability mass function (PMF), while for continuous random variables, it is represented by a probability density function (PDF).

6. Binomial Distribution

The binomial distribution is a special type of probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials. Each trial has only two outcomes: success or failure. The binomial distribution is characterized by two parameters:

1. n: The number of trials.
2. p: The probability of success in each trial.

The probability mass function for the binomial distribution is given by:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

where:

• $\binom{n}{k}$ is the binomial coefficient (number of ways to choose $k$ successes from $n$ trials).
• $p^k$ is the probability of $k$ successes.
• $(1-p)^{n-k}$ is the probability of $n-k$ failures.

For example, if you flip a coin 5 times and want to find the probability of getting exactly 3 heads, you would use the binomial distribution with $n = 5$, $k = 3$, and $p = 0.5$.

NCERT Solutions for Class 12 Maths Chapter 13

The NCERT Solutions for Chapter 13 of Class 12 Maths are designed to provide students with a comprehensive understanding of the concepts and problem-solving techniques in probability. Here is a summary of the types of problems typically included in the solutions:

1. Basic Problems on Probability: These problems involve calculating the probability of single events, both simple and compound.

2. Conditional Probability Problems: These problems require students to calculate the probability of an event given that another event has occurred, using the formula for conditional probability.

3. Multiplication Theorem Problems: These problems involve applying the multiplication theorem to find the probability of the intersection of two events.

4. Bayes' Theorem Problems: These problems involve using Bayes' Theorem to update probabilities based on new information.

5. Random Variables and Distributions Problems: These problems involve finding the probability distributions of random variables and solving related questions.

6. Binomial Distribution Problems: These problems involve using the binomial distribution to solve questions related to a fixed number of trials and probabilities of success.

Sample Problems and Solutions

Here are a few sample problems from Chapter 13 and their solutions to illustrate the types of questions students may encounter:

Problem 1: A bag contains 5 red balls and 3 green balls. Two balls are drawn randomly without replacement. Find the probability that both balls are red.

Solution:

• Total number of balls = 5 + 3 = 8.
• Number of ways to choose 2 balls out of 8 = $\binom{8}{2} = 28$.
• Number of ways to choose 2 red balls out of 5 = $\binom{5}{2} = 10$.
• Probability that both balls are red = $\frac{10}{28} = \frac{5}{14}$.

Problem 2: Given that the probability of passing an exam is 0.6 and the probability of a student studying is 0.8, find the probability that a student will pass given that they have studied, if the probability of passing given studying is 0.9.

Solution:

• Let $A$ be the event of passing and $B$ be the event of studying.
• $P(A|B) = 0.9$, $P(B) = 0.8$, $P(A \cap B) = P(A|B) \times P(B) = 0.9 \times 0.8 = 0.72$.
• $P(A) = 0.6$.
• $P(A \cap B) = P(A) \times P(B|A)$, so $P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.72}{0.6} = 1.2$. (Note: This is an indication that the given data might be incorrect, as probability cannot exceed 1.)

Problem 3: Find the probability of getting exactly 2 heads in 4 coin flips.

Solution:

• Number of trials, $n = 4$.
• Probability of success (heads), $p = 0.5$.
• Number of successes, $k = 2$.
• Probability of getting exactly 2 heads = $\binom{4}{2} (0.5)^2 (0.5)^{4-2} = 6 \times 0.25 \times 0.25 = 0.375$.

Conclusion

Chapter 13 of the Class 12 NCERT Mathematics textbook is a fundamental chapter that provides students with a solid foundation in probability. Understanding the key concepts, such as probability of events, conditional probability, multiplication theorem, Bayes' theorem, random variables, and binomial distribution, is crucial for solving a wide range of problems in probability. The NCERT Solutions for this chapter offer a comprehensive guide to mastering these concepts and preparing effectively for exams.

By practicing the problems and understanding the solutions provided, students can enhance their problem-solving skills and gain confidence in their ability to tackle probability questions in their exams and beyond.