Probability: Definition, Theorems, and Applications

Prediction is one of the important aspects of human life. Predictions are more important to find the weather forecast, to have knowledge about the calamities, the chances for our favorite team to win a match, etc. How are these predictions made? It’s possible only using probability!!!

Few centuries ago, gambling and gaming were considered to be fashionable and became widely popular among many men. As the games became more complicated, players were interested in knowing the chances of winning or losing a game from a given situation. This led to the creation of a mathematical theory of probability by two famous French Mathematicians, Blaise Pascal and Pierre de Fermat. The topic of probability is seen in many factors of the modern world. From its origin as a method of studying games, probability has been involved in a powerful and widely applicable branch of mathematics.

Probability theory in mathematics is the branch of mathematics concerned with the analysis of random phenomena. The central objects of probability theory in mathematics are random variables, and events. It provides the theoretical foundation for statistics and is used in various fields to model uncertainty and predict outcomes.

This article is about the concept of Probability class 11 and Probability class 12 including the topics Probability Formula and Rules, Theorems on Probability, Probability Distribution Functions and Applications of Probability.

Probability

Probability is the actual likelihood of any event or any situation to occur. For example, if we toss a coin the probability of getting a head is $50%$ or $\frac{1}{2}$. The application of probability is vast in many of our real-life scenarios. In games, we predict the probability of any team losing or winning, We also use probability whenever we talk about how the weather is going to be.

What is Probability?

Probability is a measure of how likely an event is to happen. With the help of probability, we can predict the chance of an event to occur. The value of probability ranges between $0$ to $1$, where $0$ represents the impossible event and $1$ represents the probability of events that are certain to happen. Now,Let us look into the basic concepts of probability.

Types of Probability

• Classical Probability: Classical probability is based on the assumption that all outcomes of a random experiment are equally likely. It is calculated as the ratio of the number of favourable outcomes to the total number of possible outcomes. For example, the probability of rolling a $3$ on a fair six-sided die is $\frac{1}{6}$.
• Empirical Probability: Empirical probability is based on observed data rather than theoretical calculations. It is determined by conducting experiments or collecting data and then calculating the probability based on the relative frequency of an event occurring in the data. For example, if a coin is flipped 100 times and lands on heads $55$ times, the empirical probability of landing on heads is $\frac{55}{100}$ = $0.55$.
• Subjective Probability: Subjective probability is based on personal judgment, intuition, or experience rather than objective data or mathematical calculation. It reflects an individual's belief about the likelihood of an event occurring, which may vary from person to person. For example, someone might estimate a $70%$ chance of their favourite sports team winning a game based on their analysis and feelings about the team's performance.
• Axiomatic Probability: Axiomatic probability is a formal framework for probability theory based on a set of axioms or fundamental principles. It provides a rigorous foundation by defining probability functions and operations through axioms, such as the probability of the entire sample space being $1$ and the additivity of mutually exclusive events.

Probability Notes

Probability notes include topics like terms used in probability, Random variables and its distributions.

Probability Class 11 notes

Terms used in Probability

• Sample Space: The set of all possible outcomes in a random experiment is called a sample space. For example, the sample space for a coin toss is {Heads, Tails}.

• Outcome and Trial: A trial means doing or performing an experiment and an outcome is the result of that experiment. For example, if rolling a dice is a trial then getting a $3$ is an outcome.
• Event: An event is some of the specific outcomes of an experiment, in other words, an event is a subset of the sample space.
• Favourable Event: Favourable events are those events associated with the experiment which are of interest to us. When rolling a dice if we are interested in even numbers then getting a $2$ is called favourable event.
• Non-favorable Event: Non-favourable events are those events associated with the experiment in which we are not interested. When rolling a die if we are interested in even numbers then getting a $3$ is called a non-favourable event.
• Exhaustive Events: Exhaustive events are all those events that take up all the possible outcomes of an experiment. When rolling a die the exhaustive events are getting a $1, 2, 3, 4, 5,$ and $6$
• Mutually Exclusive Events: When some events can not happen at the same time or simultaneously, then they are called mutually exclusive events. When tossing a coin getting head or tail are mutually exclusive events.
• Independent Events: Independent events are those events where the occurrence of one event does not affect the other event's occurrence or vice versa. For example, flipping a coin and rolling a die can be considered independent events.
• Impossible Events: An impossible event is an event which cannot happen. For example, rolling a $7$ on a standard six-sided die is an impossible event.

Probability of an Impossible event

The probability of an impossible event is always $0$, as it can not happen under any situation. If you roll a standard six-sided die, it is impossible to get a $7$, as there is no $7$ on the die. So in this case the probability of rolling a $7$ is $0$.

Probability of a sure event

The probability of a sure event is always 1 because it is certain to occur. When we flip a coin, we are certain to get a head or a tail, so getting a head or a tail in this case is a sure event. Hence, the probability of getting either heads or tails is 1, when we toss a coin.

Probability Class 12 Notes

Random Varibale

A random variable is a real valued function whose domain is the sample space of a random experiment.

Probability Distribution of a Random Variable

The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable.
The probability distribution of a random variable $X$ is the system of numbers

$
\begin{array}{rlllllll}
X & : & x_1 & x_2 & x_3 & \ldots & \ldots & x_n \\
P(X) & : & p_1 & p_2 & p_3 & \ldots & \ldots & p_n \\
& p_i \neq 0, & \sum_{i=1}^n p_i=1, & i=1,2,3, \ldots n
\end{array}
$

The real numbers $\underline{\underline{x_1},}, x_2, \ldots, x_n$ are the possible values of the random variable $X$ and $p_i(i=1,2, \ldots, n)$ is the probability of the random variable $X$ taking the value $x i$ i.e., $P\left(X=x_i\right)=p_i$

Types Of Probability Distribution:
1. Binomial distribution
2. Normal Distribution
3. Cumulative distribution frequency

Probability formulas

Probability Formulas Class 11

The probability of an event $A$ is written as $P(A)$, and it can easily calculated using the formula:

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

For example, while rolling a dice probability of getting a $6$ is $\frac{1}{6}$.

Steps to Calculate Probability

• Define the Experiment: Determine the process or action that generates outcomes.
• Identify the Sample Space: List all possible outcomes of the experiment.
• Determine the Favorable Outcomes: Identify which of these outcomes meet the criteria of the event of interest.
• Apply the Probability Formula: Use the formula to calculate the probability of the event.

Example: What is the probability of getting a number greater than $4$ when rolling a dice?

The experiment is rolling a single die.

The sample space $S$ consists of all possible outcomes: $S={1,2,3,4,5,6}$

The favourable outcomes are ${5,6}$

The required probability $= \frac{2}{6} = \frac{1}{3}$.

Probability if either of the two events occurred (Addition Rule)

The probability when either of two events occurred or the addition rule states that the probability of either of the two events occurring is the sum of their individual probabilities minus the probability of both events happening. The formula is:

$P(A \cup B) = P(A) + P(B) - P (A \cap B)$

Where $P(A \cup B)$ is the probability of either event $A$ or event $B$ occurring.

$P(A∩B)$ is the probability of both events $A$ and $B$ occurring.

$P(A)$ is the probability of event $A$ occurring.

$P(B)$ is the probability of event $B$ occurring.

Probability of Complementary event

Complementary events in probability are two such events which are mutually exclusive events which means they can not happen together, in other words, if one event occurs then the other event can not occur. The sum of the probability of two complementary events is $1$. The formula:

$P(A′) = 1 - P(A)$

where $P(A)$ is the probability of the event $A$, and $P(A′)$ is the probability of the event $A′ $ (the complement of $A$).

Probability Formulas Class 12

Conditional Probability (Probability of $B$ if $A$ has already occurred)

Conditional probability is the probability of an event $B$ occurring given that another event $A$ has already happened. It is denoted by $P(B∣A)$. Conditional probability formula is

$P(B∣A) = \frac{P(A \cap B)}{P(A)}$

Where $P(B∣A)$ is the probability of event $B$ given that $A$ has occurred.

$P(A∩B)$ is the probability of both events $A$ and $B$ occurring.

$P(A)$ is the probability of event $A$ occurring.

Probability when two events occurred simultaneously

The probability of two events occurring simultaneously has two cases, one is when the events are independent and the other one is when the events are dependent.

• For independent events: If events $A$ and $B$ are independent, the probability of both events occurring simultaneously is the product of the probability of the individual events, it is given by: $P(A∩B) = P(A) × P(B)$
• For dependent events: When events $A$ and $B$ are dependent, the probability of both events occurring simultaneously is given by: $P(A∩B) = P(A) × P(B∣A)$, where $P(B∣A)$ is the conditional probability of event $B$ given that event $A$ has already occurred.

Theorems on Probability

Theorems on probability include Bayes' Theorem, Law of total probability, and some other theorems on probability.

Theorems on Probability Class 11

Theorem 1: The sum of the probability of happening of an event and not happening of an event is equal to $1$. $P(A) + P(A') = 1$.

Theorem 2: The probability of an impossible event or the probability of an event not happening is always equal to $0$. $P(ϕ) = 0$.

Theorem 3: The probability of a sure event is always equal to $1$. $P(A) = 1$

Theorem 4: The probability of happening of any event always lies between $0$ and $1$, i.e. $0 \leq P(A) \leq 1$.

Theorems on Probability Class 12

Theorem 1: If there are two events $A$ and $B$, we can apply the formula of the union of two sets and we can derive the formula for the probability of happening of event $A$ or event $B$ as follows.

$P(A∪B) = P(A) + P(B) - P(A∩B)$

Also for two mutually exclusive events $A$ and $B$, we have $P( A U B) = P(A) + P(B)$

Bayes’ Theorem for Conditional Probability

Bayes’ Theorem provides a way to update the probability of an event based on new evidence. It is expressed as $P(A∣B) = \frac{P(B∣A). P(A)}{P(B)}$
where $P(A∣B)$ is the probability of event $A$ given that event $B$ has occurred, $P(B∣A)$ is the probability of observing $B$ given $A$, $P(A)$ is the prior probability of $A$, and $P(B)$ is the total probability of $B$

Law of Total Probability

The Law of Total Probability states that the probability of an event can be found by considering all possible ways that event can occur through a partition of the sample space. It is expressed as $P(B) = \sum_{i} P(B|A_i). P(A_i)$, where $A_i$ are mutually exclusive and exhaustive events that partition the sample space. This law allows us to compute $P(B)$ by summing the probabilities of B occurring with each $A_i$ weighted by the probability of $A_i$.

Important Points

• The probability ranges from $0$ to $1$.
• The sum of probabilities of all possible outcomes in a sample space is $1$.
• For mutually exclusive events, the probability of either event occurring is the sum of their individual probabilities.

Probability tree

A probability tree is a diagram that represents all possible outcomes of an event and their probabilities. It is useful for calculating the probabilities of combined events.

Let us see a probability tree of tossing a coin twice to have a clear understanding of this.

Application of probability

Probability has a wide variety of applications in real life. Some of the common applications that we see in our everyday lives while checking the results of the following events:

• Choosing a card from the deck of cards
• Flipping a coin
• Throwing a dice in the air
• Pulling a red ball out of a bucket of red and white balls
• Winning a lucky draw

Understanding the deck of cards

A standard deck has $52$ cards with $4$ suits in two colors [hearts, diamonds, clubs, spades] each containing $13$ ranks $(2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace)$.

Tossing of a coin, Tossing of two or more coins

Each coin toss has $2$ possible outcomes. Tossing two coins can result in $HH, HT, TH,$ or $TT$.

Rolling a dice

A standard die has $6$ faces numbered $1$ to $6$. Each face has an equal probability of $\frac{1}{6}$.

List of Topics According to NCERT/JEE MAIN

Importance of Probability

Our world is filled with uncertainty. We make decisions affected by uncertainty virtually every day. The theory of probability is used to measure this uncertianty. The uses of probability range from the determination of life insurance premium, to the prediction of election outcomes, the description of behavior of gas in a molecule, the prediction of a disaster and so on. It has been used extensively in the fields of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather Forecasting, etc.

How to Study Probability?

There are multiple ways of studying Probability. In this section, I have written the most preferred way of understanding probability.

I would recommend you to study Permutations and Combinations or Combinatorics before studying Probability or start the next section after Venn diagrams. Probability and Permutations/Combinations are related as Permutations and Combinations is a segment of mathematics concerned with counting. Do Combinatorics-based problems in probability and statistics. Now start solving probability problems with examples. Practice till you have command of this topic as it is the most basic concepts of Probability. Also, practice applications of probability in other sections of mathematics. After that, you can move to Random Variables, Conditional Probability, etc.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic. At the end of the chapter remember to make short notes on the method of solutions of different differential equations to revise quickly before exams or anytime when you are required to revise the chapter.

Best Books for Probability

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Basic Probability Theory (Robert B. Ash, Department of Mathematics, University of Illinois). Probability are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link:

• NCERT solutions for class 12 Physics.
• NCERT solutions for class 12 Chemistry.
• NCERT solutions for class 12 Mathematics.
• NCERT solutions for class 12 Biology.