DEFINITION of Conditional Probability

Conditional probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome. Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event.

BREAKING DOWN Conditional Probability

Conditional probabilities are contingent on a previous result. For example, suppose you are drawing three marbles - red, blue and green - from a bag. Each marble has an equal chance of being drawn. What is the conditional probability of drawing the red marble after already drawing the blue one? First, the probability of drawing a blue marble is about 33% because it is one possible outcome out of three. Assuming this first event occurs, there will be two marbles remaining, with each having a 50% of being drawn. So, the chance of drawing a blue marble after already drawing a red marble would be about 16.5% (33% x 50%).

Examples of Conditional Probability

As another example, suppose a student is applying for admission to a university and hopes to receive an academic scholarship. The school to which they are applying accepts 100 of every 1,000 applicants (10%) and awards academic scholarships to 10 of every 500 students who are accepted (2%). Of the scholarship recipients, 50% of them also receive university stipends for books, meals& and housing. For our ambitious student, the change of them being accepted then receiving a scholarship is .2% (.1 x .02). The chance of them being accepted, receiving the scholarship, then also receiving a stipend for books, etc. is .1% (.1 x .02 x .5).

See also, Bayes' Theorem.