Conditional Probability

Conditional Probability

If 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.

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

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

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