Probability and Statistics - JEE Main 2025

Probability and statistics are two key concepts in mathematics. Probability is based entirely on chance. Statistics, on the other hand, is involved with how we handle various data sets using various techniques. It aids in the representation of complex data in a simple and understandable manner. Nowadays, statistics are widely used in data science professions. Professionals use statistics to make business predictions. It assists them in forecasting the company's future profit or loss.

Introduction to Probability in JEE Main Maths

Probability, a crucial component in JEE Main Mathematics, serves as the mathematical lens through which uncertainties are explored and quantified. This 100-word introduction invites JEE Main aspirants to delve into the fascinating realm of probability. From basic concepts to advanced applications, probability in JEE Main Mathematics unlocks a world of predictive analytics and decision-making. As students navigate through this mathematical landscape, they unravel the intricacies of chance and randomness, preparing them to tackle real-world problems with precision. This introduction sparks the curiosity of aspiring mathematicians, setting the stage for a comprehensive exploration of probability in the JEE Main syllabus.

Important Topics of Maths Statistics and Probability Chapter

• Probability

• Probability Distribution

• Types of Events in Probability

• Statistics

• Types of Statistics

• Mean, Median and Mode

• Standard Deviation, Variance, and Mean Deviation

Maths Statistics and Probability Important Concept for JEE Main

What is Probability?

Probability refers to the possibility of any random event's outcome. For instance, when an event occurs, such as throwing a ball or selecting a card from a deck, there must be some probability associated with that event. This term refers to determining the probability of a given event occurring. 

For example, when we flip a coin in the air what are the chances of getting a head?

The number of possible outcomes determines the answer to this question. In this case, the outcome could be either head or tail. As a result, the probability of a head appearing as a result is $\dfrac{1}{2}$.

Probability All Formulas

Any event that occurs has an outcome. An event can have a variety of outcomes.

The formula for calculating probability is given by,

${P(\text{Event})} = \dfrac{\text{Total Number of Favorable Outcomes}}{\text{Total number of outcomes}}$

$P(E) = \dfrac{n(E)}{n(S)}$

Where, 

n(E) - the outcome of an event

n(S) - the total number of events in the sample space

Probability Formula for Two Events

Let A and B be two events

Probability Distribution of a Random Variate

A probability distribution describes the likelihood of different values of a random variable occurring. Common probability distributions include the normal distribution, Poisson distribution, and exponential distribution.

Types of Probability

There are four main types of probability:

• Classical Probability: This type of probability is based on the assumption that all outcomes of an event are equally likely. For example, the probability of rolling a 6 on a die is 1/6, because there is one favorable outcome (rolling a 6) and six total possible outcomes.

• Empirical Probability: This type of probability is based on the observed frequency of an event. For example, if you flip a coin 100 times and get heads 55 times, the empirical probability of getting heads is 55%.

• Subjective Probability: This type of probability is based on a person's personal beliefs about the likelihood of an event. For example, someone might believe that there is a 50% chance of rain tomorrow, based on their knowledge of the weather forecast and their own past experiences.

• Axiomatic Probability: This type of probability is based on a set of mathematical axioms. These axioms define the properties of probability and how it should be calculated. Axiomatic probability is the most rigorous type of probability, and it is the foundation of modern statistical theory.

What are the Five Probability Rules?

• The probability of an impossible event is phi or a null set.

• The sample space of an event contains the maximum probability of that event (sample space is the total number of possible outcomes).

• Any event has a probability between 0 and 1 but an event cannot have a negative probability.

• If A and B are two events that cannot occur simultaneously, then the probability of A or B occurring is the probability of A + the probability of B.

What is Statistics?

The study of data collection, analysis, interpretation, presentation, and organization is known as statistics. It is a method of gathering and summarising data. Stats are used for all data analysis, whether it is the study of the country's population or its economy.

Statistics has a wide range of applications in many fields, including sociology, psychology, geology, weather forecasting, and so on. The information gathered for analysis here could be quantitative or qualitative. Quantitative data can also be divided into two types: discrete and continuous. Continuous data has a range rather than a fixed value, whereas discrete data has a fixed value. There are many topics in statistics, and terms in statistics and formulas used in this concept.

Measures of Dispersion

Measures of dispersion are statistical tools used to gauge the degree of variation or spread in a dataset. They provide a more comprehensive understanding of data distribution, which is essential in various real-world applications.

Calculation of Mean, Median, and Mode

Before we dive into dispersion, it's crucial to understand how to calculate the mean, median, and mode.

Mean

The term mean refers to the average of a set of numbers.

$\text{Mean}=\dfrac{\text{Sum of all the numbers}}{\text{Total number of items}}$

Mean Formula is given by 

$\text{Mean} = \dfrac{\sum(f_i \cdot x_i)}{\sum f_i}$

Steps to Calculate Mean

Step 1: Find the class mark $x_i$ for each class i.e.,

$x=\dfrac{1}{2}(\text{lower limit + upper limit})$

Step 2: Calculate $f_i \cdot x_i$ for each $i$.

Step 3: Apply the formula of Mean $= \dfrac{\sum(f_i \cdot x_i)}{\sum f_i}$

Example: 

(1) Direct method of calculating mean: 4,6,8,10

The mean is calculated by adding all of the numbers and dividing the sum of numbers by the total count of numbers.

Mean $= \dfrac{4+6+8+10}{4} = \dfrac{28}{4} = 7$

(2) Calculate mean of the following numbers

Ans:

Median

The median is the central or middle value of a data set.

$\text{Median}=\dfrac{\text{Number of observations}+1}{2}$

Formula for Calculating Median:

Median $=l+h\left ( \dfrac{\left ( \dfrac{N}{2}-cf\right )}{f} \right )$

Example: 

(1) Direct method of calculating Median 1,2,4,5,7,8,10,12,13,15,16,18,20

Manually, here the median obtained is 10 as it is the middle term from the data set.

Using formula, $\text{Median}=\dfrac{\text{Number of observations}+1}{2}$

$\text{Median}=\dfrac{13+1}{2}=7$

Hence median = 10

(2) Calculate median from the given table below:

Ans: 1st calculate the cumulative frequency that is, by adding the sum of each frequency in a frequency distribution table to its predecessors

As we have to find the median then, $N = \dfrac{N}{2} = \dfrac{80}{2} = 40$

The cumulative frequency just above 40 is 58, and the associated class is 24-32.

Therefore, l = 24, h = 8, f = 24, cf = c.f. of preceding class = 34, and $\dfrac{N}{2}=40$.

Hence, Median $=l+h\left ( \dfrac{\left ( \dfrac{N}{2}-cf\right )}{f} \right )$

$\Rightarrow  24+8 \dfrac{(40 – 34)}{24} = 26$ 

Mode

The most frequently occurring data item is mode.

Example: (20,20,21,21,21,22,22,23,23,23,23,23,23,24,25,26)

In the given data set, the mode is obtained by selecting the most frequently occurring item in the data set.

Hence the mode here is 23.

Note: The relationship between the mean, median, and mode is

Mode = 3(Median) – 2(Mean)

Grouped and Ungrouped Data

Statistics can be applied to both grouped and ungrouped data.

• Ungrouped Data: When data is given in its raw form with individual values, measures of dispersion are calculated directly using the formulas mentioned above.

• Grouped Data: In real-life scenarios, data is often grouped into intervals or classes. In such cases, we use the midpoint of each class and the frequency to calculate the mean, median, and mode.

Calculation of Standard Deviation, Variance, and Mean Deviation

Once you have calculated the mean, you can move on to calculating standard deviation, variance, and mean deviation to gain insights into the data's spread.

Standard Deviation: Standard deviation (σ) measures the average distance of data points from the mean. A higher standard deviation indicates greater variability. The formula for standard deviation is:

Standard Deviation Formula

$\sigma = \sqrt{\frac{\sum(x - \mu)^2}{n}}$

Where 'x' represents data points, 'μ' is the mean, and 'n' is the number of data points.

Variance: Variance (σ^2) is the square of the standard deviation and represents the average of the squared differences between data points and the mean. The formula for variance is:

Variance Formula

$\sigma^2 = \frac{\sum(x - \mu)^2}{n}$

Mean Deviation: Mean deviation provides information about the average deviation of data points from the mean. It is calculated using the formula:

Mean Deviation Formula

$\text{Mean Deviation} = \frac{\sum |x - \mu|}{n}$

Statistics Formulas for JEE Main

Statistics Formulas for JEE Main, provide a concise and comprehensive overview of key statistical principles and equations, equipping you with the tools necessary for success in JEE Main examinations. Whether you are exploring measures of central tendency, probability distributions, or hypothesis testing, this guide serves as a valuable resource to strengthen your statistical foundations and excel in your JEE Main studies.

In statistics, there are five key formulas. They are:

1. $\text{Mean}=\bar{x}=\dfrac{\sum x_{i}}{N}$

Where, $x_{i}$ - terms in the data set

$N$ - the total number of terms

2. If ‘n’ is odd number, $\text{Median}=\left(\dfrac{n+1}{2}\right)^{th}$

If ‘n’ is an even number, $\text{Median} = \dfrac{1}{2}\left\{\left(\dfrac{n}{2}\right)^{th}\text{ variable } + \left(\dfrac{n}{2} + 1\right)^{th}\text{ variable}\right\}$

3. The most frequently occurring value in a data set is referred to as the mode.

4. Variance $=\sigma^{2}=\dfrac{\sum\left(x_{i}-x\right)^{2}}{N}$

5. Standard Deviation $=\sqrt{\text { Variance }}$

Or, $\sqrt{\dfrac{\sum\left(x_{i}-x\right)^{2}}{N}}$

Measures of Central Value

Sometimes when we are working with large amounts of data, we need one single value to represent the whole set. In math, there are three measures to find the central value of a given set of data. They are

Example: 

(1) Direct method of calculating mean: 4,6,8,10

The mean is calculated by adding all of the numbers and dividing the sum of numbers by the total count of numbers.

Mean $= \dfrac{4+6+8+10}{4} = \dfrac{28}{4} = 7$

(2) Calculate mean of the following numbers

Ans:

Median

The median is the central or middle value of a data set.

$\text{Median}=\dfrac{\text{Number of observations}+1}{2}$

Formula for Calculating Median:

Median $=l+h\left ( \dfrac{\left ( \dfrac{N}{2}-cf\right )}{f} \right )$

Example: 

(1) Direct method of calculating Median 1,2,4,5,7,8,10,12,13,15,16,18,20

Manually, here the median obtained is 10 as it is the middle term from the data set.

Using formula, $\text{Median}=\dfrac{\text{Number of observations}+1}{2}$

$\text{Median}=\dfrac{13+1}{2}=7$

Hence median = 10

(2) Calculate median from the given table below:

Ans: 1st calculate the cumulative frequency that is, by adding the sum of each frequency in a frequency distribution table to its predecessors

As we have to find the median then, $N = \dfrac{N}{2} = \dfrac{80}{2} = 40$

The cumulative frequency just above 40 is 58, and the associated class is 24-32.

Therefore, l = 24, h = 8, f = 24, cf = c.f. of preceding class = 34, and $\dfrac{N}{2}=40$.

Hence, Median $=l+h\left ( \dfrac{\left ( \dfrac{N}{2}-cf\right )}{f} \right )$

$\Rightarrow  24+8 \dfrac{(40 – 34)}{24} = 26$ 

Mode

The most frequently occurring data item is mode.

Example: (20,20,21,21,21,22,22,23,23,23,23,23,23,24,25,26

In the given data set, the mode is obtained by selecting the most frequently occurring item in the data set.

Hence the mode here is 23.

Note: The relationship between the mean, median, and mode is

Mode = 3(Median) – 2(Mean)

Concept of Probability in Statistics

Understanding probability helps us understand statistics and how to apply it because data used in statistical analyses often contains some amount of random variation.

Probability and statistics are in fact closely entwined linked and there is importance of probability in statitstics . 

For example, when an anthropologist studies a small group of people from a larger population, the results of his research will include some random variations that he must account for using statistics.

Application of Probability

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 modeling.

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

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

• In the share market, chances of getting the hike of share prices.

Application of Statistics

Statistics has numerous applications in mathematics as well as in everyday life. The following are some examples of statistics applications:

• Applied statistics, theoretical statistics, and mathematical statistics are all types of statistics.

• Data mining and machine learning

• In society, statistics

• Statistical analysis

• The application of statistics to the mathematics of the arts

Probability Notes for JEE Main 

In the journey of mastering JEE Main Mathematics, a solid grasp of probability is essential, and the comprehensive Probability Notes by Vedantu serve as a key to unlocking that understanding. Tailored for JEE Main aspirants, Probability notes meticulously cover fundamental concepts, probability distributions, and their applications. With a focus on clarity and precision, the notes guide students through the intricacies of probability theory, preparing them for the challenges of the JEE Main exam. Packed with illustrative examples and problem-solving strategies, these notes are a valuable resource, empowering students to confidently navigate the probability-based questions that await them in the examination hall. You can access the Probability notes from Vedantu’s website.

Statistics and Probability Examples:

Example 1:  Find the mean, and mode of the following data

2, 4, 7, 5, 10, 7, 12, 6, 4, 3.

Ans: Given data is 2, 4, 7, 5, 10, 7, 12, 6, 4, 3. 

Total elements in the data is = 10

Sum of all the numbers = 2 + 4 + 7 + 5 + 10 + 7 + 12 + 6 + 4 + 3 = 60

$\text{Mean }= \dfrac{\text{Sum of all the numbers}}{\text{Total number of items}}$

$\text{Mean }= \dfrac{60}{10} = 6$

Mode = 7 as it is occurring 2 times.

Example 2: Five blue, four green, and five red balls are stored in a bucket. Manooj is asked to pick two balls at random from the bucket without replacing them, and then to pick one more ball. What is the possibility he picked 2 green balls and 1 blue ball?

Ans: Five blue, four green, and five red balls gives a total count of balls = 14

The probability of drawing 1 green ball from the four $=\dfrac{4}{14}$

As it is asked to pick 2 green balls so the probability of drawing another green ball $= \dfrac{3}{13}$

Probability of drawing 1 blue ball $= \dfrac{5}{12}$

Therefore probability of picking 2 green balls and 1 blue ball $= \dfrac{4}{14} \times \dfrac{3}{13} \times  \dfrac{5}{12} $

$\Rightarrow \dfrac{2}{7}\times \dfrac{3}{13}\times \dfrac{5}{12}$

$\Rightarrow \dfrac{2\times 3\times 5}{7\times 13\times 2\times 6}$

$\Rightarrow \dfrac{5}{182}$

Solved problems of Previous Year Question

1. One coin is thrown 100 times. What is the probability of getting a tail as an odd number?

Ans: Let us consider p = Probability of getting tail $= \dfrac{1}{2}$

q = Probability of getting head $= \dfrac{1}{2}$

As the coin is thrown 100 times, n = 100, and p + q = 1

Let X be an event of getting tail

So, the required probability $= P (X = 1) + P (X = 3) +….. + P (X = 99)$

$\Rightarrow {}^{100} {C}_{1}\left(\dfrac{1}{3}\right)\left(\dfrac{1}{2}\right)^{99}+{ }^{100} {C}_{3}\left(\dfrac{1}{2}\right)^{100}+\cdots+{ }^{100} {C}_{99}\left(\dfrac{1}{2}\right)^{100}$

$\Rightarrow \dfrac{{ }^{100} {C}_{1}+{ }^{100} {C}_{3}+\cdots+{ }^{100} {C}_{99}}{2^{100}}$

$\Rightarrow \dfrac{\dfrac{2^{100}}{2}}{2^{100}}$

$\Rightarrow \dfrac{2^{99}}{2^{100}} = \dfrac{1}{2}$

2. If a variable takes the discrete values $\alpha − 4, \alpha − \dfrac{7}{2}, \alpha − \dfrac{5}{2}, \alpha − 3, \alpha − 2, \alpha + \dfrac{1}{2}, \alpha − \dfrac{1}{2}, \alpha + 5$ where $(\alpha > 0)$, then the median is ____________.

Ans: Given values are $\alpha − 4, \alpha − \dfrac{7}{2}, \alpha − \dfrac{5}{2}, \alpha − 3, \alpha − 2, \alpha + \dfrac{1}{2}, \alpha − \dfrac{1}{2}, \alpha + 5$

On rearrange the data in ascending order, we get $\alpha − \dfrac{7}{2}, \alpha − 3, \alpha − \dfrac{5}{2}, \alpha − 2, \alpha − \dfrac{1}{2}, \alpha + \dfrac{1}{2}, \alpha − 4, \alpha + 5$

Median $= \dfrac{1}{2} \left[\text{value of 4th item }+\text{ value of 5th item}\right]$

Median $= \dfrac{(\alpha − 2)+\dfrac{\alpha − 1}{2}}{2}$

$\Rightarrow \dfrac{2\alpha − \dfrac{5}{2}}{2}$

$\Rightarrow \alpha − \dfrac{5}{4}$

3. If A and B are two independent events such that $P (A \cap B^\prime) = \dfrac{3}{25}$ and $P(A^\prime \cap B) = \dfrac{8}{25}$, then $P(A) =$ _______.

Ans: Since the two events are independent.

Then, $P (A \cap B^\prime) =  P (A) \times P (B^\prime) = \dfrac{3}{25}$

$\Rightarrow P (A) \times (1 − 2 P (B)) = \dfrac{3}{25}$ ….(i)

Similarly, $P (B) \times (1 − P (A)) = \dfrac{8}{25}$ ….(ii)

Solving equation (i) and (ii), we get

$P (A) = \dfrac{1}{5} \text{ and } \dfrac{3}{5}$

Practice Problems

1. Find the mean, median, and mode of the following data.

6,10,8,10,10,7,11,12,15

Ans: Mean=10

Median=10

Mode = 10

2. If one card is drawn at random from a deck of 52 cards, what are the chances that it will be a king or a spade?

Ans: $\dfrac{4}{13}$

Hint: Divide king and spade with total deck of cards and calculate the probability of king and spade by, P (King or spade) = P (king) + P (spade)− P (King and spade)

JEE Main Maths - Statistics and Probability Study Materials

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JEE Main Maths Chapters 2025

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Conclusion

Probability and statistics is a scoring topic in mathematics, with questions that are very basic and simple. Probability and statistics has an overall weightage of 11 to 15 marks, with main questions coming from topics in statistics such as Conditional Probability, Properties of Conditional Probability, Multiplication Theorem, Bayes Theorem, and Probability Distributions, Theorem of Total Probability. Success in JEE Main, as well as future academic and professional pursuits, hinges on understanding these concepts. This chapter offers students a valuable skill set for interpreting and critically evaluating data, making predictions, and solving problems in various scientific and technical domains. It underscores the practical significance of statistics and probability in our lives and highlights their role in shaping the future of science and technology.

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