## Conditional Probability

### Conditional probability is a concept in probability theory that quantifies the likelihood of an event occurring given that another event has already occurred

Conditional probability is a concept in probability theory that quantifies the likelihood of an event occurring given that another event has already occurred. In other words, it measures the probability of an event A happening, given that event B has occurred.

The conditional probability of event A given event B is denoted as P(A|B) and is calculated as the probability of both events A and B occurring together (P(A ∩ B)) divided by the probability of event B occurring (P(B)):

P(A|B) = P(A ∩ B) / P(B)

To understand this concept better, let’s consider a simple example:

Suppose we are interested in the probability of a student passing an exam (event A), given that they have studied for it (event B). Let’s assume that 60% of the students who study pass the exam, and the overall pass rate is 40%.

Here, event A is passing the exam, and event B is studying for the exam. We want to find the conditional probability of passing the exam given studying.

P(A) = 0.40 (Overall pass rate)

P(B) = 0.60 (Probability of studying)

P(A ∩ B) = 0.60 (Probability of both passing and studying)

Using the formula, we can calculate the conditional probability:

P(A|B) = P(A ∩ B) / P(B)

= 0.60 / 0.60

= 1

This means that if a student has studied for the exam, the conditional probability of them passing is 1, which implies that it is guaranteed they will pass.

Conditional probability is a valuable tool in many fields, including statistics, machine learning, and decision-making processes. It helps quantify the uncertainty associated with events based on the information available.

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