*A*and

*B*are two events in a random experiment, then the conditional probability of the event

*A*, once event

*B*has occurred, is the ratio of the probability that

*A*and

*B*occur simultaneously to the probability of

*B*(considered to be non-zero here).

### Notation

The way to note the “conditional probability of event A once event B has occurred” is \(\textrm{P}_\textrm{B}(\textrm{A})\) which is read as: “the probability of A based on B.”

### Formula

The conditional probability of event A, once event B has occurred, is given by:

\(\textrm{P}_\textrm{B}(\textrm{A}) = \dfrac{\textrm{P}(\textrm{A} ∩ \textrm{B})}{\textrm{P}(\textrm{B})}\)

### Example

Consider a random experiment that consists of randomly drawing one card from a deck of 52 cards.

Consider events A: drawing a king and B: drawing a red card.

We know that there are 4 kings in the deck and 26 red cards, including two red kings.

\(\textrm{P(B)}\) = \(\dfrac{26}{52}\) = \(\dfrac{1}{2}\)

\(\textrm{P(A ∩ B)}\) = \(\dfrac{2}{52}\) = \(\dfrac{1}{26}\)

\(\textrm{P}_\textrm{B}(\textrm{A})\) = \(\dfrac{\dfrac{1}{26}}{\dfrac{1}{2}}\) = \(\dfrac{1}{13}\)