Probability and Statistics

Probability is a measure of understanding how likely an event is to occur. It is represented on a scale from 0 and 1, where 0 denotes that the chance of an event occurring is impossible. If an event has a probability of 1, it indicates a sure event. We can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes. In mathematics and statistics, probability helps in forecasting outcomes and making informed decisions.  

In mathematics, statistics is a subfield that provides methods to draw conclusions about populations based on sample data. By collecting, analyzing, interpreting, presenting, and organizing data, researchers can make informed choices and meaningful decisions. The methods included in statistics are inferential statistics to test hypotheses and descriptive statistics to summarize data. 
 

Here is a list of words that are associated with the mathematical disciplines such as probability and statistics: 

• Random experiment: The precise outcome of a random experiment is unpredictable. An experiment is a process involving a series of steps that produce observable outcomes.
  
   
• Outcome: It is the possible results of an experiment. Sample space refers to any possible result in a collection of results and is denoted as ‘S’. 
  
   
• Sample space: In a given experiment, the complete set of all possible outcomes is known as sample space. For example, heads and tails are the sample space for a fair coin flip.
  
   
• Event: Any area of the sample space that consists of specific outcomes is called an event. For instance, if event A takes place, it indicates that one of the outcomes in A must have happened. If event A is rolling an even number on a fair six-sided die, then receiving 2, 4, or 6 indicates that event A took place. If the received numbers are 1, 3, or 5, it indicates that event A did not happen.  
  
   
• Trial: Tossing a coin once is considered a trial. It is a single execution of an experiment or test. 
  
   
• Mean: It is the average of all possible values of a random variable in a random experiment.  
  
   
• Expected value: It is the weighted average of all possible outcomes of a random variable. Each outcome of a random variable is weighted by its probability. For instance, if we roll a fair six-sided die, the expected value is the average of all possible outcomes, which is 3.5.  
   

Formulas of probability and statistics help us to solve complex mathematical problems easily, and they aid in making well-informed decisions and conclusions. Here are some of the common formulas of probability and statistics: 

Probability formulas:

Probability is a measure used to calculate the likelihood of an event occurring. The formula for calculating probability is: 

P(A) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Here, P(A) is the probability of an event A happening.

Favorable outcomes are the cases where event A happens. 

The total number of possible outcomes is the total number of results.

The probability of an event that is certain is 1. The probability of an event that is impossible to happen is 0. So, the values of probability always lie between 1 and 0. Probability can be written in a percentage format by multiplying the given value by 100. 

For instance, the probability of getting heads when tossing a fair coin is:

P (Heads) = 1 / 2

A fair coin has two sides, and only one of them is a heads:

P (Heads) = 1 / 2 = 0.5 = 50%

Additional rule formula:  

To calculate the probability that at least one of two mutually exclusive occurrences will occur, we can use the addition rule of probability. In the formula for mutually exclusive events with no overlap, the likelihood of either event A or B happening is calculated by adding their probabilities separately. If A and B are mutually exclusive events: 

(A or B) = P(A ∪ B) = P(A) + P(B) 

For non-mutually exclusive events with overlapping, the formula is: 

P(A or B) = P(A ∪ B) = P(A) + P(B) - P(A ∩ B) 

Multiplication rule formula: 

We can use this formula to calculate the probability of two independent events happening together. The probability of both events happening, if A and B are dependent on one another, is equal to the product of the probability of A and the conditional probability of B given that A has happened.  

P(A ∩ B) = P(A) × P(B∣A)

Here, P(B∣A) is the probability of B happening after A has already happened. 

Bayes’ rule:

This method is used to update probabilities when new information is available. It determines the likelihood that event A will occur given the occurrence of event B. 

P(A∣B)= P(B∣A) × P(A) /  P(B)

Here, P(A | B) is the probability of A happening given that B has occurred.

P(B | A) is the probability of B happening given that A has occurred. 

P(A) and P(B) are the individual probabilities of A and B.
 

Other important rules related to probability are given below: 

• The probability is between 0 and 1: If an event cannot happen, then its probability is 0. If an event is sure to happen, its probability is 1. 
  
   
• The sum of all probabilities is 1: The sum of all possible outcomes equals 1 or 100%. 
  
   
• Complement rule: An event's likelihood of occurring (P(A)) plus its likelihood of not occurring (P(not A)) = 1. One common way to write P(not A) is 1−P(A). 
  
   

Statistics formulas:

Here are some of the common formulas for statistics listed below: 

Mean: It is the average of a set of given numbers of data. The formula for calculating the mean is: 

Mean = Sum of all values / Total number of values

To find the mean, we first need to add all the numbers together and then divide it by the total number of values. 

Median:

The middle number or value in an arranged dataset is known as the median. If the given data consists of odd numbers, the median will be the middle value. If the given numbers are even, the median will be the average of the two middle values.  

The formula for finding the median for odd numbers is:

Median = Value at (n +1 / 2)th position

The formula for finding the median for even numbers is:

Median = 1 / 2 ( Value at n/2 th position + Value at (n/2 + 1)th position

Here n is the number of values in the given dataset. 

Mode:

It is the most frequently appearing value in a dataset. A dataset can have one mode (unimodal), multimodal, or no mode at all. 

Variance:

It is a measure that explains how far the values are from the mean in a given dataset. To calculate the variance, first, we need to find the difference between each value and its mean. Then, square the differences and find the average of these squared deviations. The formula for calculating variance is:

 ∑(Each value−Mean)² / Total number of values 

Or 

σ² = ∑(xi- x̄)² / N

Here, σ² is the variance

∑(xi- x̄)² is the sum of squared deviations

N is the total number of terms. 

Standard deviation:

It is the square root of the variance. It shows how much the values spread out from the mean. The formula for calculating standard deviation is: 

Standard Deviation = √Variance

 
Or

√σ² = √∑(xi- x̄)² / N

Here, xi is the each value in the dataset.

√σ² is the standard deviation

x̄ is the mean 
 

Numerous topics are covered under probability and statistics, which help analyze and predict outcomes.
Events in Probability are categorized into: 

• Simple event: When there is just one possible outcome, a simple event occurs. For instance, while tossing a coin, getting heads is one simple event, whereas getting tails is another simple event. The probability of a simple event can be calculated using the formula: 
  
  P(Simple Event) = 1 / Total Possible Outcomes
  
   
• Compound Event: Two or more simple events combine to form a compound event. For example, flipping a coin twice and getting heads both times is a compound event. Multiplying the probabilities of the individual simple events gives the probability of a compound event:
  
  P(Compound Event) = P(Event 1) × P(Event 2)

• Independent Event: Independent events occur when the results of one event do not influence those of another. For example, when you throw a coin more than one time, each flip remains independent of the previous ones. 

• Dependent events: When one event influences the probability of another, this is known as a dependent event. For instance, drawing a marble from a bag without replacing it,  may change the probability of drawing subsequent marbles.

• Complementary Event: An event’s complement (denoted as A) includes all results that were not a part of event A. If rolling an even number on a six-sided die is event A, then rolling an odd number is its complement. The probability is given by: 

            P(Not A) = 1−P(A)
 

Probability Distribution:

A probability distribution explains how probabilities are allocated to various possible values of a random variable. To understand the probability of different outcomes, the probability distribution is helpful. There are two main types of probability distributions: 

• Discrete Probability Distribution: Deals with countable quantities, such as rolling a die.
• Continuous Probability Distribution: Uncountable quantities, like weight or height, are involved in this distribution.  

Probability Functions:

A mathematical framework for defining probability distributions is offered by probability functions. The two main categories of probability functions are the Probability Mass Function (PMF) and the Probability Density Function (PDF). For discrete variables, PDF is used and for continuous variables, PDF is utilized.
 

Among the most important subjects in statistics are:

Descriptive statistics: 

They use graphical representations and numerical measures to meaningfully summarize and arrange data.  It is a branch of statistics that uses summary statistics to provide a clear understanding of the data. 

Measures of Central Tendency:

It helps identify a typical value in a dataset. The methods of central tendency are:
Mean: By adding up all of the data points and dividing by the total number, the mean value is determined.
Median: The middle number in an ordered set of data is called the median.
Mode: The value that appears the most frequently in the dataset is called the mode.

Measures of Variability:

Variability demonstrates the degree of dispersion of data values.
Standard deviation: Values' deviation from the mean is measured by the standard deviation.
Variance: The average squared departure from the mean is known as variance.

Inferential Statistics:

Based on sample data, inferential statistics enable inferences about the population. It is impractical to get data from a whole population. Rather, we make generalizations using inferential methods. For instance, a sample can be used to determine the overall average rather than asking every high school student in a country about their performance on any entrance exam. 

Data Representations:

Representing data effectively helps in analysis and interpretation. Common methods include:

Graphical Representations

• Pie Charts 
• Line Graphs.
• Bar Graphs 
• Scatter Plots 
• Frequency Distribution Tables 
• Box-and-Whisker Plots (Box plots)
• Dot Plots 
• Pictograms

Sampling Techniques:

To ensure accuracy and representativeness, sampling procedures assist in choosing a subset of a population for analysis.  Here are some common sampling techniques:

• Simple Random Sampling
• Stratified Sampling
• Systematic Sampling
• Cluster Sampling
• Convenience Sampling
• Quota Sampling
• Purposive Sampling
• Snowball Sampling
   

To make informed decisions based on data and uncertainty, probability and statistics are helpful. In various fields, such as health care, finance, economics, statistics, and technology these two essential concepts are utilized. 

• In the field of stock market analysis and marketing, to forecast stock prices and evaluate risks, investors employ probability models. Also, most of the companies use statistical data to forecast future sales patterns.
  
   
• Using probability, researchers examine how diseases spread and how well treatments work. Additionally, it is used to examine the efficiency of new medications.
  
   
• To ensure the quality of their products, businesses employ statistical process control measurements. To test the reliability of machinery, engineers use probability distributions to examine machine failure rates.
  
   
• Algorithms use probability to classify data and make predictions. In the technology sector, statistical analysis helps detect threats, such as identifying fraudulent banking transactions or malware in cybersecurity.