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# RS Aggarwal Class 12 Chapter 29 Solutions (Probability)

RS Aggarwal Solutions for Class 12 Chapter 29 Probability prepares you to tackle all the complex problems of this chapter. You will learn deeply about the chances of occurrence of a given event using various advanced theorems. This is a chapter of RS Aggarwal Solutions which will help you move a step ahead of the concepts of Probability that you learned in your previous classes. The knowledge of Probability is essential not only from the examination perspective but the chapter of Probability is also necessary for all the social sciences and other scientific studies.

There are a total of 22 questions in 2 exercises in this chapter of RS Aggarwal. Various concepts such as conditional Probability, Multiplication Theorem of Probability, Bayes’ Theorem, Partition of Sample Space, Theorem of Total Probability etc are all comprehensively covered in a wide variety of questions in both the exercises. These questions are strictly based on the latest CBSE pattern of question papers. Also, the solutions prepared at Instasolv are as upgraded as the questions. Probability has a substantial amount of weightage in the pan India competitive entrance exam, therefore it is very important that you grasp it with robust practice.

We, at Instasolv, have intended to bring absolute clarity in the concepts of the chapter Probability of RS Aggarwal Class 12 Maths Solutions by writing detailed steps with reasons in the answers provided by Instasolv. The maths faculties of Instasolv keep updating itself from time to time in regard to the latest CBSE guidelines and are therefore committed to providing one of the most reliable supplementary resources.

## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 29 – Probability

Introduction:

In this chapter, we will be discussing the concept of the Conditional probability of an event on the condition that another event has occurred, which will be helpful in understanding the Bayes’ theorem, multiplication rule of Probability and independent events and their study.

Conditional Probability:

If E and F are two events linked with the same sample space of a random experiment, the conditional probability of the event E given that F has transpired, i.e. P (E|F) is given by

P(E|F) = P(E ∩ F) / P(F),    provided P(F) ≠ 0

Multiplication Theorem:

In conditional Probability, we know that the probability of occurrence of some event is affected when some of the possible events have already occurred. When we know that a particular event B has occurred, then instead of S, we concentrate on B for calculating the probability of occurrence of event A given B.

P(E ∩ F) = P(E) P(F|E) = P(F) P(E|F) ,  provided P(E) ≠ 0 and P(F) ≠ 0.

This result is called the multiplication rule of Probability.

Multiplication rule of Probability for three events:

If E, F, and G are the three events, we can write,

P(E ∩ F ∩ G) = P(E) P(F | E) P(G | (E ∩ F)) = P(E) P(F | E) P(G | EF).

In the same way, the multiplication rule of Probability can be broadened for four or more events.

Independent Events:

The meaning of Independent Events is that events that occur without the interference of each other. The events are independent of each other, i.e., the occurrence of one event does not affect the occurrence of the other event. The Probability of occurring of the two events are independent of each other.

An event A is said to be independent of any other event B if and only if the probability of occurrence of one of them is independent of the occurrence of the other event.

Suppose if we draw two cards from a pack of cards one after the other. The results of the two draws are independent if the cards are drawn with replacement i.e., the first card is put back into the pack before the second draw. If the cards are not replaced then the events of drawing the cards are not independent.

An event A is said to be independent of any other event B if the conditional Probability of A is equal to the unconditional Probability of A. P(B) ≠ 0.

P(A | B) = P(A)

Bayes Theorem:

Here we take E1, E2, E3, … , En as mutually disjoint events whose P(Ei) ≠ 0, (i = 1, 2, …, n), then for any random event A which is a subset(proper or improper) of the union of events Ei such that P(A) > 0, we have

E1, E2, E3, … , En represents the partition of the sample space S.

Important Highlights of the Chapter:

1. In conditional Probability, you are taught, if there are two events from one sample space then what would be the effect of the occurrence of one event on the occurrence of the other event.
2. In the multiplication theory, we learn about the occurrence of two or more events.
3. Using Bayes’ theorem, we can find reverse Probability using conditional Probability.

### Exercise Wise Discussion of RS Aggarwal Solutions for Class 12 Chapter 29 Probability

1. The exercise solutions of Probability have topic-wise coverage.
2. The initial exercises consist of comparatively simpler events to calculate the Probability of such as rolling a die or tossing a coin.
3. The later exercise solutions contain mostly practical problems which are more practical in nature and are very important for all the exams.
4. By solving these questions, you will be able to manage your time accordingly in the exam hall.

### Benefits of RS Aggarwal Solutions for Class 12 Maths Chapter 29 by Instasolv

1. The subject matter experts at Instasolv have made the complex Probability problems the simplest ones with their expertise. It is highly recommended that you practise these solutions for clarity in the concepts.
2. The answers by Instasolv will become your go-to revision resource once you achieve your desired goal of obtaining full marks.
3. Instasolv has covered each and every question to consolidate the solutions of all the frequently asked queries.
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