Statistics For Programmers - Conditional Probability

The conditional probability of an event is the probability of an event occuring given that another event has occured. The It is denoted as $P(B|A)$ and is read as "the probability of B given A". That is the probability of event B occuring knowing that event A has occured.

The formula for conditional probability is

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

Where (P(A \cap B)) is the probability of both events A and B occuring.

Let's break this down with an example. Given a standard 52 card deck of playing cards, we want to find the probability of drawing 2 aces in a row. We will be drawing cards without replacing them - this is an important detail.

The probability $P(A)$ is the probability of the first event (drawing an ace).

\[
P(A) = \frac{4}{52} = \frac{1}{13}
\]

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