Conditional probability

When we compute the probability of event $F$ assuming that the event $E$ has already occurred, we call this the conditional probability of $F$ given $E$.

We denote this probability as $P(F|E)$. We read $P(F|E)$ as “the probability of $F$ given that $E$ has occurred”, or in a quicker way, “the probability of $F$ given $E$.”

If $E$ and $F$ are dependent events in a sample space with equally likely outcomes, then:

If $E$ and $F$ are dependent events in a sample space, then

This formula can also be rearranged algebraically for $P(E⋂F)$ by multiplying both sides by $P(E)$.

Events $E$ and $F$ are independent events if $P(E|F) = P(F)$. In other words, if $P(E|F) ≠ P(F)$, then $E$ and $F$ are dependent.