# Conditional Probability Calculator

Instructions: Use this Conditional Probability calculator to compute the conditional probability $$\Pr(A | B)$$. Please provide the probability $$\Pr(A \cap B)$$ and $$\Pr(B)$$ in the form below:

Please indicate the value of $$\Pr(A \cap B)$$ =
Please indicate the value of $$\Pr(B)$$ =

The concept of conditional probability is one of the most crucial ideas in Probability and Statistics. And it is a quite simple idea: The conditional probability of an event $$A$$ given an event $$B$$ is the probability that $$A$$ happens under the assumption that $$B$$ happens as well.

This is, we restrict the sample space to outputs in which $$B$$ happens, and we look for the probability that $$A$$ occurs in that subset sample space.

### So, what is the formula for conditional probability?

In mathematical terms, the conditional probability $$\Pr(A|B)$$ is computed using the following formula:

$\Pr(A|B) = \displaystyle \frac{\Pr(A \cap B)}{\Pr(B)}$

The above expression can be rewritten and it also provides a way to compute the probability of the intersection of two event, when the conditional probability is known:

$\Pr(A \cap B) = \Pr(A|B) \Pr(B)$

### Why is conditional probability important?

The concept of conditional probability is crucial because it represents the fact of real life that as we know more information about some event, we can refine our idea of the likelihood of an event. This idea of computing a probability given that we know that certain even is true is a representation of how our brain works, and hence, make the idea of conditional probability very important.

Also, the concept of conditional probability and the law of multiplication play a crucial role for the construction of the Total Probability Rule as well as Bayes' Theorem.

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