Probability and Statistics
2026-02-28 11:07 Diff
https://brightchamps.com/en-us/math/data/probability-and-statistics

Formulas of probability and statistics help us 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) = \(\frac{1}{2}\)


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


P(Heads) \(=\frac{1}{2} = 0.5 = 50%\)

Addition Rule Formula 


To calculate the probability that at least one of the two mutually exclusive events 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, then: 


(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. 

\(P(A∣B) = \frac{P(B∣A) × P(A)}{P(B)}\)

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).