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 [latex]\textrm{P}_\textrm{B}(\textrm{A})[/latex] 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:[latex]\textrm{P}_\textrm{B}(\textrm{A}) = \dfrac{\textrm{P}(\textrm{A} ∩ \textrm{B})}{\textrm{P}(\textrm{B})}[/latex]
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.[latex]\textrm{P(B)}[/latex] = [latex]\dfrac{26}{52}[/latex] = [latex]\dfrac{1}{2}[/latex]
[latex]\textrm{P(A ∩ B)}[/latex] = [latex]\dfrac{2}{52}[/latex] = [latex]\dfrac{1}{26}[/latex]
[latex]\textrm{P}_\textrm{B}(\textrm{A})[/latex] = [latex]\dfrac{\dfrac{1}{26}}{\dfrac{1}{2}}[/latex] = [latex]\dfrac{1}{13}[/latex]