Probability and Statistics

Probability And Statistics are the two important concepts in Maths. Probability is all about chance. Whereas statistics is more about how we handle various data using different techniques.  It helps to represent complicated data in a very easy and understandable way. Statistics and probability are usually introduced in Class 10, Class 11 and Class 12 students are preparing for school exams and competitive examinations. The introduction of these fundamentals is briefly given in your academic books and notes. The statistic has a huge application nowadays in data science professions. The professionals use the stats and do the predictions of the business. It helps them to predict the future profit or loss attained by the company.

What Is Probability and Statistics?

Probability is a concept used in math and science to know the likelihood or occurrence of an event. For example, when a coin is tossed, there is a probability to get a head or tail.

Statistics deals with a set of data. Sometimes we may be interested in finding the most favorite or frequently used item from a set of data.

In such cases, we can work on the data set to make an analysis and conclusion. 

This concept that deals with data analysis, interpretation, and presentation of data in a more meaningful way is statistics.

We will discuss the probability and statistics formula in the sections below to find out the values for various real-world situations.

What is Probability?

Probability denotes the possibility of the outcome of any random event. The meaning of this term is to check the extent to which any event is likely to happen. For example, when we flip a coin in the air, what is the possibility of getting a head? The answer to this question is based on the number of possible outcomes. Here the possibility is either head or tail will be the outcome. So, the probability of a head to come as a result is 1/2.

The probability is the measure of the likelihood of an event to happen. It measures the certainty of the event. The formula for probability is given by;

P(E) = Number of Favourable Outcomes/Number of total outcomes

P(E) = n(E)/n(S)

Here,

n(E) = Number of event favourable to event E

n(S) = Total number of outcomes

What is Statistics?

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. It is a method of collecting and summarising the data. This has many applications from a small scale to large scale. Whether it is the study of the population of the country or its economy, stats are used for all such data analysis.

Statistics has a huge scope in many fields such as sociology, psychology, geology, weather forecasting, etc. The data collected here for analysis could be quantitative or qualitative. Quantitative data are also of two types such as: discrete and continuous. Discrete data has a fixed value whereas continuous data is not a fixed data but has a range. There are many terms and formulas used in this concept. See the below table to understand them.

Terms Used in Probability and Statistics

There are various terms utilised in the probability and statistics concepts, Such as:

• Random Experiment
• Sample Sample
• Random variables
• Expected Value
• Independence
• Variance
• Mean

Let us discuss these terms one by one.

Random Experiment

An experiment whose result cannot be predicted, until it is noticed is called a random experiment. For example, when we throw a dice randomly, the result is uncertain to us. We can get any output between 1 to 6. Hence, this experiment is random.

Sample Space

A sample space is the set of all possible results or outcomes of a random experiment. Suppose, if we have thrown a dice, randomly, then the sample space for this experiment will be all possible outcomes of throwing a dice, such as;

Sample Space = { 1,2,3,4,5,6}

Random Variables

The variables which denote the possible outcomes of a random experiment are called random variables. They are of two types:

1. Discrete Random Variables
2. Continuous Random Variables

Discrete random variables take only those distinct values which are countable. Whereas continuous random variables could take an infinite number of possible values.

Independent Event

When the probability of occurrence of one event has no impact on the probability of another event, then both the events are termed as independent of each other. For example, if you flip a coin and at the same time you throw a dice, the probability of getting a ‘head’ is independent of the probability of getting a 6 in dice.

Mean

Mean of a random variable is the average of the random values of the possible outcomes of a random experiment. In simple terms, it is the expectation of the possible outcomes of the random experiment, repeated again and again or n number of times. It is also called the expectation of a random variable.

Expected Value

Expected value is the mean of a random variable. It is the assumed value which is considered for a random experiment.  It is also called expectation, mathematical expectation or first moment. For example, if we roll a dice having six faces, then the expected value will be the average value of all the possible outcomes, i.e. 3.5.

Variance

Basically, the variance tells us how the values of the random variable are spread around the mean value. It specifies the distribution of the sample space across the mean.

List of Probability Topics

Basic probability topics are:

List of Statistical Topics

Basic Statistics topics are:

Probability and Statistics Formulas

Probability Formulas: For two events A and B:

Statistics Formulas : Some important formulas are listed below:

Let x be an item given and n is the total number of items.

What Are the Three Types of Probability?

• Theoretical or Classical Probability
• Experimental Probability
• Axiomatic Probability

Theoretical or Classical Probability 

Theoretical probabilty measures the favorable outcome of an event.

Experimental Probability

Experimental probability measures the total number of favorable outcomes for the number of times an experiment is repeated. 

Axiomatic Probability

Axiomatic probability is one more way to describe the outcomes of an event.

There are three rules or axioms which apply to all types of probability.

These rules were defined by Kolmogorov and is called Kolmogorov’s axioms.

The three axioms are as follows:

• For any event, the probability is greater than or equal to 0.
• Sample space defines the set of all possible outcomes of an event.
• If A and B are two mutually exclusive outcomes (two events that cannot occur at the same time) then the probability of A or B occurring is a probability of A plus the probability of B.

What Are the Five Rules of Probability?

• The probability of an impossible event is phi or a null set.
• The maximum probability of an event is its sample space (sample space is the total number of possible outcomes).
• The probability of any event exists between 0 and 1. 0 can also be a probability.
• There cannot be a negative probability for an event.
• If A and B are two mutually exclusive outcomes (two events that cannot occur at the same time) then the probability of A or B occurring is the probability of A plus the probability of B.

Scroll down to find some probability and statistics formulas.

What Is the Formula to Calculate Probability?

Any event that happens has an outcome. There may be so many outcomes for an event. 

The formula to calculate probability is as follows.

Let P(X) denote the probability of an event ‘X’.

What Is the Formula for Statistics?

There are five important formulas used in statistics. They are as follows.

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

• Mean 
• Median
• Mode

Mean

Mean represents an average of a given set of numbers.

For example in the given data set,3,5,7,9

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

Measures of Spread

While measuring a central value, we are given a data set. To know how wide is the data set, we use measures of spread.

They also give us a better picture if the calculated central value (Mean, median, or mode) correctly depicts the set of values.

For example,

Here we have the marks obtained by students in a test

25,40,55,60,70,86,90,100

While calculating the average marks obtained by students in a 100 marks test, by calculating the mean, we can only find the average marks obtained by the students, but we do not know how spread the marks are from 0 to 100. 

While calculating the average marks obtained by students in a  100 marks test, the mean score cannot be more than Here, the mode is 100. 

In such cases, measures of spread are useful.

The most-used measures of spread are

• Range
• Quartiles and InterQuartiles Range
• Standard Deviation 
• Variance

Range 

Range represents the difference between the minimum and maximum values in a data set.

Quartile

As the name suggests, a quartile is a measure of spread that groups a given set of values in quarters.

To use this measure,

We first arrange the values in an increasing order.

Make four equal groups of the values.

For example, in the data set shown below,

Number of values = 7

Interquartile

The interquartile range is obtained by subtracting the third quartile (Q3) and the first quartile (Q1). It is a place where most of the values lie in a data set or we can say sometimes the interquartile range and the mean value are the same.

For example, 

Variance

• Variance is a measure that gives us a approximate idea about the spread of data. It is not very accurate. 
• The use of variance value is it can be used to calculate the standard deviation of a data set. 

Standard Deviation

• Standard deviation is a measure that tells us how far is a data value from the mean. 
• It is obtained by taking the square root of variance.

For example, 

Let us say the average score to pass a test is 60. The score obtained by a student is 75

. This means that the student is well above the average marks. 

Comparing Data

In statistics, the following concepts are used to compare data

• Univariate and Bivariate data
• Scatter
• Outliers
• Correlation

Univariate Data

Univariate data involves comparing one type of data. 

For example, we can compare the heights of students 

Heights in feet and inches: 5 feet 9 inches, 5 feet 7 inches, 5 feet 5 inches, 5 feet 10 inches.

Bivariate Data

Bivariate data involves comparing two types of data.

For example, the number of shirts sold on all the days of a week. Here sales and days of the week are two types of data. 

Scatter Plots

• Scatter plots are used to know how one data relates to another. 
• A series of points represent the scatter plots.

For example, a graph representing the age and weight in pounds of a person is given. 

We can measure how the weight increases as the age increases. 

Outlier

A value that lies out of range in a given data set is called an outlier. In other words, they are the smallest and the largest extreme values.

For example,

Here,  1 and  650

are extreme values which do not relate to the other values in the list.

Correlation

When two sets of data values increase or decrease together we say that they are correlated. 

Correlation is positive if the two data sets increase together.

For example,

Electricity consumption increases as the number of people increases.

Correlation is negative if one value decreases as the other value increases.

For example, 

The distance to cover decreases as you walk for more time.

Important Notes

1. Outcome is the result of an event. Sample space is the total number of possible outcomes of an event.
2. Value of probability always lies between 0 and 1.
3. Central value of a set of data can be measured using mean, median, or mode.

Now let us discuss some probability and statistics examples.

Solved Examples

Here are some examples based on the concepts of statistics and probability to understand better. Students can practice more questions based on these solved examples to excel in the topic. Also, make use of the formulas given in this article in the above section to solve problems based on them.

Example 1: Find the mean and mode of the following data: 2, 3, 5, 6, 10, 6, 12, 6, 3, 4.

Solution:

Total Count: 10

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

Mean = (sum of all the numbers)/(Total number of items)

Mean = 60/10 = 6

Again, Number 6 is occurring for 3 times, therefore Mode = 6. Answer

Example 2: A bucket contains 5 blue, 4 green and 5 red balls. Sudheer is asked to pick 2 balls randomly from the bucket without replacement and then one more ball is to be picked. What is the probability he picked 2 green balls and 1 blue ball?

Solution: Total number of balls = 14

Probability of drawing

1 green ball = 4/14

another green ball = 3/13

1 blue ball = 5/12

Probability of picking 2 green balls and 1 blue ball = 4/14 * 3/13 * 5/12 = 5/182.

Example 3: What is the probability that Ram will choose a marble at random and that it is not black if the bowl contains 3 red, 2 black and 5 green marbles.

Solution: Total number of marble = 10

Red and Green marbles = 8

Find the number of marbles that are not black and divide by the total number of marbles.

So P(not black) = (number of red or green marbles)/(total number of marbles)

= 8 /10

= 4/5

Example 4: Find the mean of the following data:

55, 36, 95, 73, 60, 42, 25, 78, 75, 62

Solution: Given,

55 36 95 73 60 42 25 78 75 62

Sum of observations = 55 + 36 + 95 + 73 + 60 + 42 + 25 + 78 + 75 + 62 = 601

Number of observations = 10

Mean = 601/10 = 60.1

Example 5: Find the median and mode of the following marks (out of 10) obtained by 20 students:

4, 6, 5, 9, 3, 2, 7, 7, 6, 5, 4, 9, 10, 10, 3, 4, 7, 6, 9, 9

Solution: Given,

4, 6, 5, 9, 3, 2, 7, 7, 6, 5, 4, 9, 10, 10, 3, 4, 7, 6, 9, 9

Ascending order: 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 9, 9, 9, 9, 10, 10

Number of observations = n = 20

Median = (10th  + 11th observation)/2

= (6 + 6)/2

= 6

Most frequent observations = 9

Hence, the mode is 9.

Solved Examples

Alan rolled a dice. What is the probability of getting number  3?

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

 7,10,9,10,10,8,12,13,16

Solution

Let us arrange the numbers in increasing order.

 7,8,9,10,10,10,12,13,16

Find the range, quartile for the following set of numbers.

10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60

Solution

 Range:

FAQs on Probability and Statistics

1. What are random variables?

The possible outcomes from a random event are called random variables.

2. Explain binomial distribution. 

The probability of success or failure of an event is defined in a binomial distribution. For example, in a test, there is a probability of passing or failing. 

3. Explain normal distribution.

When there are different sets of data (high or low or mixed) we need an approximate value of the given probability distribution. In that case, we use a normal distribution. It is represented using a bell curve.

4. What is sample space?

The set of all possible outcomes or results of a particular experiment or an event is called sample space.

5. What are the measures of spread in probability?

Range, quartiles, standard deviation, and variance are the measures of spread in probability.

6. What is the formula to find the probability of an event?

The probability of an event is the ratio of a number of favorable outcomes to the total number of favorable outcomes of the event.

7. What is the range of the value of probability?

The value of probability always lies between 0 and 1.

8. What are the measures of central tendency?

Mean, median, and mode are the measures of central tendency.

9. Name some of the fields that use statistics?

Business, economics, banking, and weather forecasting are some of the fields using statistics.

10. What is the difference between probability and statistics?

Probability deals with the likelihood of events whereas statistics deals with analysis and interpretation of frequency of events.