### 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.