# Conditional Probability

### [Conditional Probability](https://en.wikipedia.org/wiki/Conditional_probability)

This is defined as the probability of an event occurring, assuming that one or more other events have already occurred. Two events, $$A$$ and $$B$$ are considered to be *independent* if event $$A$$ has no effect on the probability of event $$B$$ (i.e. $$P(B \vert A) = P(B)$$). If events $$A$$ and $$B$$ are *not independent*, then we must consider the probability that both events occur. This can be referred to as the *intersection* of events $$A$$ and $$B$$, defined as $$P(A \cap B) = P(B \vert A) \* P(A)$$. We can then use this definition to find the *conditional probability* by dividing the probability of the intersection of the two events $$(A \cap B)$$ by the probability of the event that is assumed to have already occurred (event $$A$$):

$$\large P(B \vert A)=\frac{P(A \cap B)}{P(A)}$$

### [Bayes' Theorem](https://en.wikipedia.org/wiki/Bayes'_theorem)

Let $$A$$ and $$B$$ be two events such that $$P(A \vert B)$$ denotes the probability of the occurrence of $$A$$ given that $$B$$ has occurred and $$P(B \vert A)$$ denotes the probability of the occurrence of  $$B$$given that $$A$$ has occurred, then:

$$\large P(A \vert B) = \frac{P(B \vert A) \* P(A)}{P(B)} = \frac{P(B \vert A) \* P(A)}{P( B \vert A) \* P(A) + P(B \vert A^c) \* P(A^c)}$$

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