Bayes’ Theorem, also known as Bayes Rule or Bayes Law states the probability that one event is based on the occurrence of some other events. It depends upon the concepts of conditional probability. In simple terms, Baye’s theorem determines the conditional probability of an event X given that event Y has already occurred, The probability of some events can be determined by this theorem depending upon some events.

As we know that the possibility of heart disease increases with increasing age. Therefore, if we know someone’s age and know the probability of getting heart disease, we can easily find the opportunity of the person having a heart disease.

Suppose, We have been given a bag of blue, red, and purple balls. It’s been said that we have to pick a blue ball only after picking a purple ball. This is sure that we can only pick green balls after picking a red ball. This makes the case of conditional probability.

## Baye’s Theorem Formula

Baye’s theorem formula is given as:

P (M ∣ N) =

Where:

- M, N are events
- P (M ∣ N) = Probability of M given N is true.
- P (N ∣ M) = Probability of N given M is true.
- P(M), P (N) = The independent probabilities of M and N.

## What is the Conditional Probability in Bayes Theorem?

The conditional probability states the probability of an event X that is based on the occurrence of another event Y. We derived the Baye’s theorem from the definition of conditional probability. There are two conditional probability in Baye’s theorem.

## Important Notes on Baye’s Theorem

- We can determine conditional probability using Baye’s theorem.
- When two events X and Y are independent, P(X ∣ Y) = P(X) and P(Y ∣ X) = P (Y).
- We can calculate conditional probability for continuous random variables using Baye’s theorem.

# Probability

## What Is Probability?

Probability is nothing but a possibility of a particular event taking place. Probability generally lies between 0 and 1. The 0 in the probability indicates the impossibility of the event and on the other hand, 1 represents the surety of a particular event. Probability can be defined as the (%) of a particular event. The only difference in probability is it cannot be 0% or 100%. This is because 0% means the particular event is impossible to occur whereas 100% means the surety of a particular event to occur.

## Application of Probability In Real-Life

Probability is used in different fields including:

** **

**Weather:**Probability is used by Meteorologists to predict how likely an event has to occur which includes snow, rain, or other weather.**Sports:**Probability is used in sports to estimate the possibility of winning or losing a particular team. These are generally used by coaches or athletes.

### ● **Insurance:** Probability is used in insurance to estimate how likely a particular car or any other thing would get insurance.

### Types of Events in Probability

### ● **Impossible and Sure events: **The probability 0 of a particular event indicates an impossible event. On the other hand, probability 1 of a particular event indicates a sure event.

### ● **Simple events: **In the simple event, there is only a single point of sample space. For example, the probability of getting 5 when dies is tossed.

### ● **Compound events: **In probability, when there is more than one possible outcome, then it is termed a compound event.

### ● **Independent event and dependent event: **When an occurrence of a particular event is dependent on the occurrence of the other event, then the event is termed as a dependent event. When the occurrence of any event is independent of the occurrence of the other event, then the event is termed an independent event.