Conditional Probability

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

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:

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