# Conditional Probability Formula

## Conditional Probability Formula

The potential of an event or outcome occurring based on the existence of a prior event or outcome is known as Conditional Probability. It is determined by multiplying the likelihood of the earlier occurrence by the increased likelihood of the later, or conditional, event. This is where the independent event and dependent event notion is used. Consider a student who misses class twice each week, omitting Sunday. What are the possibilities that he will take a leave of absence on Saturday the same week if it is known that he will be absent from school on Tuesday? It has been noted that situations where the outcome of one event influences the outcome of a subsequent event are termed as Conditional Probability.

## What Is Conditional Probability Formula?

One of the core concepts in probability theory is the concept of the Conditional Probability Formula. The Conditional Probability Formula calculates the likelihood of an event, say B, given the occurrence of another event, say A.

By knowing the Conditional Probability Formula of event B given that event A has occurred and the individual probabilities of events A and B, the Bayes theorem may be used to calculate the Conditional Probability Formula likelihood of event A given that event B has occurred.

The Conditional Probability Formula of P(A | B) is undefinable if P(B)=0. (Event B did not take place).

## Formula for Conditional Probability

The Conditional Probability Formula is as follows:

P(A and B)/P = P(A | B) (B)

Another way to write it is,

P(A|B)=P(A∩B)P(B)

### Derivation of Conditional Probability Formula

Based on the existence of a prior event or outcome, the Conditional Probability Formula is the likelihood that a future event or outcome will occur. The probability multiplication rule serves as the basis for the Conditional Probability Formula.

P(A) = Chance that event A will occur

P(B) is the likelihood that event B will occur.

P(AB) suggests that both occurrences, A and B, have taken place or at least some of their common components.

Every outcome that is not contained in B but is in A is removed if B has also occurred, narrowing the sample space needed to determine B.

The only way that A can occur is when the outcome belongs to the set AB, since the set of possible outcomes for A and B is thus limited to those in which B occurs. As a result, we divide P(A B) by P(B), which is equivalent to limiting the sample space to those instances where B occurs.

### Application of Conditional Probability Formula

The Conditional Probability Formula is frequently used to anticipate the results of actions like tossing dice, picking a card from a deck, and flipping a coin. Additionally, it aids in the analysis of the given data set by data scientists, improving results. Creating more precise prediction models is helpful for machine learning developers.

Examples that have been solved to understand the Conditional Probability Formula.

Example 1: Of a group of ten persons, four purchased apples, three purchased oranges, and two purchased both apples and oranges. Using the Conditional Probability Formula, what is the likelihood that a consumer who selected apples at random also purchased oranges?

Solution:

Let those who purchased apples be A and those who purchased oranges be O.

It follows that

P(A) = 4/10, 40%, or 0.4.

P(O) = 3/10, 30%, or 0.3.

Hence,

P(AO) = 2/10, or 20%, or 0.2

Using the Conditional Probability Formula,

50% is equal to P(O|A) = P(AO) / P(A) = 0.2 / 0.4 / 0.5

Given that they also purchased apples, there is a 50% chance that the customer also purchased oranges.

### Examples Using Conditional Probability Formula

Consider yourself a furniture salesperson. On any given day, 30% of new customers to your business are likely to buy a couch. However, the likelihood may be 70% if they visit your store in the month before the Super Bowl. The Conditional Probability Formula of selling a couch in the month before the Super Bowl may be expressed as P (Selling a couch | Super Bowl month), where the symbol | stands for “given that”. This Conditional Probability Formula gives us a mechanism to define probabilities when our opinions about the likelihood that one event will occur (in this case, the sale of couches) given that another event has occurred change (in this case, the advent of the month preceding the Super Bowl).

### 1. 3 coins are in a piggy bank. 1 fake-headed coin and 2 regular coins are included. {P (H) = 1 P (H)=1}. A. A person chose a coin at random and tossed it. How likely is it that it will bring up heads?

Based on the supposition,

Assume that A1A1 is the requirement that you choose a regular coin and A2A2 is the requirement that you choose the 2-headed coin. A1A1 and A2A2 create a sample space partition, so keep that in mind.

Now,

P(H|A1) = 0.5 and P(H|A1) = 0.5

P H|A2 = 1 P H|A2 = 1

Here, using the probability principle, we write

P H P H = P H | A1 P H | A2 P H | A1 – P H | A2 P ‘A1’ plus P ‘H|A2’ ‘A2’

1/2.2/3 + 1.1 /3 = 2/3

So the probability obtained is 2/3.