Conditional Probability - Probability Given Prior Knowledge

Conditional probability is the probability of an event occurring given that another event has already occurred. It is one of the most important and widely applied ideas in the whole of probability, used in medicine, spam filters, weather forecasting, and legal reasoning.

The Reverend Thomas Bayes, an English statistician and Presbyterian minister, developed the theorem that extends conditional probability into one of mathematics' most powerful tools for updating beliefs with new evidence – but he never published it himself. It was found among his papers after his death and published in 1763 by his friend Richard Price under the title An Essay towards solving a Problem in the Doctrine of Chances. Bayes' theorem, built directly on the P(B | A) formula taught here, now drives spam filters that update the probability an email is junk as each new word is scanned, and medical diagnosis tools that combine a test result with how common a disease already is in the population.

Notation and Formula

P(B | A) is read as "the probability of B given A". It means: assuming A has already happened, what is the probability that B also happens?

P(B | A) = P(A and B) ÷ P(A)

Rearranging: P(A and B) = P(A) × P(B | A)

This second form is called the General Multiplication Rule and works for both independent and dependent events.

When Is P(B | A) = P(B)?

If A and B are independent, knowing A occurred gives no information about B, so P(B | A) = P(B). This is actually the mathematical definition of independence.

Worked Examples

A bag has 4 red and 6 blue balls. One ball is drawn and NOT replaced. A second ball is then drawn. Find P(red on second | red on first).

After one red is removed: 3 red remain out of 9 total.
P(red 2nd | red 1st) = 3/9 = 1/3.

P(red 1st and red 2nd) = P(red 1st) × P(red 2nd | red 1st) = 4/10 × 3/9 = 12/90 = 2/15.

From a two-way table: 40 students; 18 boys and 22 girls. 10 boys and 14 girls play sport. A student is chosen at random. Find P(plays sport | boy).

Given the student is a boy (18 boys total), 10 of them play sport.
P(plays sport | boy) = 10/18 = 5/9 ≈ 0.556.

P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. Find P(A | B) and P(B | A).

P(A | B) = P(A and B) ÷ P(B) = 0.2 ÷ 0.4 = 0.5. Since P(A | B) = P(A), A and B are independent.
P(B | A) = 0.2 ÷ 0.5 = 0.4. Since P(B | A) = P(B), confirmed independent.

Real-World Application: Medical Testing

Suppose a disease affects 1% of a population and a test for it is 99% accurate. Even with a positive result, the probability of actually having the disease (conditional probability) may be much lower than you expect – this is a famous result known as the base rate fallacy, and conditional probability is the tool that reveals the truth.

Key Takeaways

P(B | A) = P(A and B) ÷ P(A) – the probability of B given A has occurred.
General multiplication rule: P(A and B) = P(A) × P(B | A).
If P(B | A) = P(B), then A and B are independent.
Conditional probability is essential for dependent events and real-world reasoning under uncertainty.

Practice: Conditional Probability

Conditional Probability Formula