Probability is the science of determining the likelihood of events occurring. It is concerned with the roll of a dice or the fall of cards in a game at its most basic level. Probability, on the other hand, is essential to science and life in general.

Probability is employed in various applications, including weather forecasting and calculating the cost of your insurance premiums.

 Probability is a metric for determining the possibility of an event occurring. Many things are impossible to forecast with 100% accuracy. Using it, we can only anticipate the probability of an event occurring, i.e. how probable it is to occur. Probability can range from 0 to 1, 0 indicating an improbable event and 1 indicating a certain event. Probability for Class 10 is an important topic for students because it teaches all of the fundamental concepts. All of the events in a sample space have the same probability.

You have no idea what will happen when you roll a dice. This is an example of a random experiment. A random experiment, in particular, is a method of observing something unknown.

The outcome of the random experiment is known after the experiment. A random experiment's result is called an outcome.

The sample space is the collection of all conceivable outcomes. As a result, the sample space is our universal set in the context of a random experiment.

A set of experiment outcomes can be defined as a probability event. In other words, in probability, an event is a subset of the sample space.

What exactly is sample space?

A random experiment's sample space (or individual space) contains the whole range of possible outcomes. Probability refers to the possibility of an event occurring. Any event has a probability of occurring between 0 and 1.

Types of events :

• • • Impossible and Sure Events.

    • Simple Events.

    • Compound Events.

    • Independent and Dependent Events.

    • Mutually Exclusive Events.

    • Exhaustive Events.

    • Complementary Events.

    • Events Associated with "OR".

    • Events Associated with "AND".

    • Event E1 but not E2.

In probability, the algebra of events consists of the following:     

• • • Complimentary event.

    • The 'A or B' Event.

    • The 'A and B' Event.     

    • The 'A but not B' Event.

Mutually exclusive events

Two events are mutually exclusive or disjoint in probability theory if they do not occur simultaneously. A good example is the set of outcomes of a single coin toss, which can finish in either heads or tails.

but not both. Both outcomes are collectively exhaustive while tossing the coin, implying that at least one of the outcomes must occur. Hence these two options together exhaust all possibilities. However, not all mutually exclusive events are mutually exclusive in the same way. When we toss six-sided dice, for example, the outcomes 1 and 4 are mutually exclusive (both 1 and 4 cannot come up as a result simultaneously) but not collectively exhaustive.

Exhaustive events

Exhaustive events occur when a sample space is partitioned into several mutually exclusive events, the union of which constitutes the sample space itself. A collectively exhaustive event is a collection of all potential elementary events for a given experiment. 

A set of events is either jointly or collectively exhaustive in probability theory and logic if at least one of the events must occur for certain. For example, when rolling unbiased six-sided dice, the outcomes 1, 2, 3, 4, 5, and 6 are all exhaustive. Because the outcomes cover the whole range of possible outcomes for an experiment, we can affirm this. In the same way, a coin flip can result in either heads or tails. As a result, they are both exhaustive events because they can occur during an experiment.

We are more familiar with the word 'chance' than the word 'probability' in our everyday lives. Because mathematics is all about measuring things, probability theory measures the possibilities of events occurring or not occurring. In probability, there are various sorts of events. We'll take a closer look at the definition and criteria of axiomatic probability in this section. We address random experiments, sample space, and other events related to the many experiments in the traditional approach to probability.

1. Probability of an event

The probability of an event can only be 1 or 0. It can neither be in fractions nor points. The event will either happen or will not.

     2. Probabilities of equally likely outcomes

If all of the sample space outcomes have the same likelihood of occurring, they are said to be equally likely. It is difficult to determine whether the results are equally likely. Still, for this lesson, we will assume that the outcomes are equally likely in the majority of the experiments.

For example: like tossing a coin.

     3. Probability of the event 'A or B'

The likelihood of A or B is determined by whether or not mutually exclusive events (those that cannot occur simultaneously) exist.

Disjoint events are those in which two events A and B are mutually exclusive. The likelihood of two discontinuous events, A and B, occurring is:

p(A or B) = p(A) + p(B). 

    4. Probability of event 'not A'

The likelihood of an event not occurring is equal to 1 minus the probability of the event occurring. If knowing that one of two events, A and B, occurs does not affect the probability of the other occurring, they are said to be independent.

Question 1: Calculate the probability of rolling a die three times and obtaining three.

Solution:

S = {1, 2, 3, 4, 5, 6,}

n(S) = 6 = total number of outcomes

Let A be the probability of getting three.

n(A) = 1 = number of favourable outcomes

A = {3}

P(A) = n(A)/n(S) = 1/6 Probability

As a result, P(getting 3 on a die roll)

= 1/6.

Question2: If two dice are rolled, what is the probability that the sum will be: equivalent to four?

Solution:

To calculate the chance that the sum is 4, we must first compute the sample space S of two dice, as illustrated below.

S = (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6) (3,1),(3,2),(3,3),(3,4),(3,5),(3,6)

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6) (5,1),(5,2),(5,3),(5,4),(5,5),

As a result, n(S) = 36.

Let A be the probability of the sum of

the numbers on the dice equaling four.

There are three alternative results, each of which adds up to four, as follows:

A = (1,3), (2,2), and (3,1)

P(A) = n(A) / n(S) = 3 / 36 = 1 /12.

Q 1. What Is Conditional Probability?

The likelihood of an event occurring if another event has already occurred is known as conditional probability. One of the most fundamental notions in probability theory is the concept of probability distributions. It's important to note that conditional probability does not imply that there is always a causal relationship between the two events, nor that they happen at the same time.

Q 2. Are the exercises given in the Ch 16 Class 11 Maths important?

Yes, it is required of you to do so. These questions will help you bridge your knowledge gap and give you an idea of which ideas you need to focus on more. Furthermore, the final paper and the way it is set will be similar to previous exercises, so you will be comfortable with the exam.

Q 3. What is a sample space and how does it help?

According to probability theory, the sample space (also known as sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment.

A random experiment's sample space is the collection of all conceivable results. A subset of the sample space is an event associated with a random experiment. Any outcome's probability is a number between 0 and 1. All of the outcomes have a probability of one.