RS Aggarwal Solutions Class 8 Chapter-25 Probability (Ex 25B) Exercise 25.2

Probability is a measure of how likely an event is to occur. Many events are impossible to predict with absolute certainty. Using it, we can only predict the likelihood of an event occurring, i.e. how likely it is to occur. Probability can range from 0 to 1, with 0 indicating an impossible event and 1 indicating a certain event. Probability for Class 10 is an important topic for students because it explains all of the fundamental concepts of this topic. The Probability of all events in a sample space equals one.

For example, when we toss a coin, we can get either Head OR Tail; there are only two possible outcomes (H, T). However, if we toss two coins into the air, there are three possible outcomes: both coins show heads, both show tails, or one shows heads and one tail, i.e. (H, H), (H, T) (T, T).

Probability Formula

According to the Probability formula, the likelihood of an event occurring is equal to the ratio of the number of favourable outcomes to the total number of outcomes.

The likelihood that an event will occur P(E) = the number of favourable outcomes divided by the total number of outcomes.

Students frequently confuse "favorable outcome" with "desirable outcome." This is the fundamental formula. However, there are some additional formulas for various situations or events.

Tree of Probability

The tree diagram aids in organizing and visualizing the various possible outcomes. The branches and the tree's ends are the two most important positions. Each branch's Probability is written on the branch, and the ends contain the final outcome. When determining when to multiply and when to add, tree diagrams are used.

Probability Types

Probabilities are Classified into three types:

• Probability Theoretical

• Probability of Experiment

• Probability Axiomatic

• Probability Theoretical

It is based on the likelihood of something occurring. The reasoning behind Probability is the foundation of theoretical Probability. If a coin is tossed, the theoretical Probability of getting ahead is 1/2.

Probability of Experiment

It is founded on the results of an experiment. The experimental Probability can be calculated by dividing the total number of trials by the number of possible outcomes. For example, if a coin is tossed ten times and the head is recorded six times, the experimental Probability of heads is 6/10, or 3/5.

Probability Axiomatic

A set of rules or axioms that apply to all types is established in axiomatic Probability. Kolmogorov established these axioms, which are known as Kolmogorov's three axioms. The axiomatic approach to Probability quantifies the chances of events occurring or not occurring. This concept is covered in detail in the axiomatic Probability lesson, which includes Kolmogorov's three rules (axioms) as well as various examples.

The likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome is known as conditional Probability.

The Event's Probability

Assume that an event E can occur in r of n probable or possibly equally likely ways. The event's likelihood of occurring or success is then expressed as;

P(E) = r/n

The likelihood that the event will not occur, also known as its failure, is expressed as follows:

P(E') = (n-r)/n  = 1-(r/n) 

E' denotes that the event will not take place.

As a result, we can now say;

P(E) + P(E') = 1.

This means that the sum of all probabilities in any random test or experiment is 1.

What are the Events That are Equally Likely?

When two events have the same theoretical Probability of occurring, they are referred to as equally likely events. The outcomes of a sample space are said to be equally likely if they all have the same chance of occurring. For example, if you roll a die, the chance of getting 1 is 1/6. Similarly, the likelihood of getting all of the numbers from 2,3,4,5, and 6 at the same time is 1/6. As a result, the following are some examples of equally likely outcomes when rolling a die:

Getting a 3 and a 5 on a die throw

Obtaining an even and an odd number on a die

Because the probabilities of each event are equal, rolling a die and getting a 1, 2, or 3 are all equally likely events.

Events That Aren't Related

The possibility that there will be only two outcomes, stating whether or not an event will occur. Examples of complementary events include a person coming or not coming to your house, getting a job or not getting a job, and so on. Essentially, the complement of an event occurs in the exact opposite Probability that it will not occur. Here are some more examples:

• Today will either rain or not rain.

• The student will either pass or fail the exam.

• You either win or you don't.

Theoretical Probability

The first work on Probability theory, The Book on Games of Chance, was written in the 16th century by J. Cardan, an Italian Mathematician, and Physician. Knowledge of Probability has drawn the attention of great Mathematicians since its inception. Thus, Probability theory is the branch of Mathematics concerned with the possibility of events occurring. Although there are many different interpretations of Probability, Probability theory precisely interprets the concept by expressing it through a set of axioms or hypotheses. These hypotheses contribute to the Probability in terms of a possibility space, which allows for a measure with values ranging from 0 to 1. This is referred to as the Probability measure, and it refers to a set of possible outcomes of the sample space.

Probability Density Function (PDF)

The Probability Density Function (PDF) is a Probability function that represents the density of a continuous random variable that falls within a given range of values. The normal distribution and how mean and deviation exist are explained by the Probability Density Function. The standard normal distribution is used to create a database or statistics, which are frequently used in science to represent real-valued variables with unknown distributions.

Probability Applications

In real life, Probability has a wide range of applications. Some of the common applications we see in our daily lives when checking the outcomes of the following events:

• Selecting a card from a deck of cards

• Tossing a coin

• Tossing a dice into the air

• Taking a red ball from a bucket full of red and white balls

• Taking part in a lucky draw

Other Important Probability Applications

• It is used in a variety of industries for risk assessment and modelling.

• Weather forecasting, also known as weather prediction, is the prediction of changes in the weather.

• The likelihood of a team winning a sport based on its players and overall strength.

• In the stock market, there is a chance that share prices will rise.