Conditional Probability

Conditional probability is the probability of some event $A$ , given the occurrence of some other event $B$ . This is denoted $P\left(A\text{\hspace{0.17em}}|\text{\hspace{0.17em}}B\right)$ and is read “the probability of $A$ , given $B$ ”. In other words, we are calculating probabilities conditional on knowing further information partway through the experiment.

$P\left(A\text{\hspace{0.17em}}|\text{\hspace{0.17em}}B\right)=\frac{P\left(A\text{\hspace{0.17em}}\cap \text{\hspace{0.17em}}B\right)}{P\left(B\right)}$

Example:

A math teacher gave her class two tests. $30%$ of the class passed both tests and $45%$ of the class passed the first test. What percent of those who passed the first test also passed the second test?

Two-thirds or approximately $66.7%$ of the class passed the second test.