http://www.ck12.org Chapter 3. An Introduction to Probability
3.4 Conditional Probability
Learning Objectives
- Calculate the conditional probability that eventAoccurs, given that eventBoccurs.
Sometimes, we wish to change the probability of an event when we are bound to certain conditions. For example,
we know that the probability of observing an even number on a throw of a die is 0.5 (simple eventA). However,
suppose that we throw the die and the result is a number that is 3 or less (simple eventB). Would the probability of
observing an even number on that particular throw still be 0.5? The answer is no because with the introduction of
the eventB, we have reduced our sample space from 6 simple events to 3 simple events. In other words, with the
introduction of a particular condition (the eventB) we have changed the probability of a particular outcome. The
Venn diagram below shows the reduced sample space for this experiment given that eventBhas occurred.
The only even number in the sample spaceBis the number 2. We conclude that the probability thatAoccurs, given
thatBoccurs is 1 : 3, or 1/3. We denote it by the symbolP(A|B), which reads "the probability ofA, givenB". So
for the die toss experiment, we write
P(A|B) =
1
3
.
Conditional Probability of Two Events
Definition Conditional Probability
IfAandBare two events, then the probability of the eventAto occur,giventhat eventBoccurs is called a
conditional probability. We denote it by the symbolP(A|B), which reads "the probability ofAgivenB."