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Conditional Probability

Conditional probability is the probability of some event $A$ , given the occurrence of some other event $B$ . This is denoted $P (A | B)$ 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.

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

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?

$\begin{array}{l} P (Second | First) = \frac{P (First and Second)}{P (First)} \\ = \frac{0.30}{0.45} \\ = \frac{2}{3} \\ = 66.7 % \end{array}$

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