Algebra 2 · Unit 11 · Topic 11.4
Conditional Probability and Independence
Conditional probability is P(A | B) = P(A and B)/P(B) when P(B) > 0.
Definition and explanation
Conditional probability is P(A | B) = P(A and B)/P(B) when P(B) > 0. Events A and B are independent when P(A and B) = P(A)P(B); mutually exclusive events are not generally independent.
Worked example
Among 40 students, 16 play an instrument, 10 play a sport, and 4 do both. Given that a student plays a sport, the probability they play an instrument is 4/10 = 0.4. This equals the overall instrument probability 16/40 = 0.4, so these two events are independent in this group.
Common mistake
The denominator of a conditional probability is the size or probability of the conditioning event, not the whole group.
Key takeaway
Conditional probability is P(A | B) = P(A and B)/P(B) when P(B) > 0.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.