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Conditional Probability and Bayes' Theorem, with a Worked Test Example

How conditional probability works, how to reverse it with Bayes' theorem, and why a test that catches 95% of faults can still be wrong most of the time.

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Key takeaways

  • P(A | B), the probability of A given B, is P(A and B) divided by P(B): restrict attention to the outcomes where B happened.
  • Events are independent when P(A | B) = P(A), equivalently P(A and B) = P(A)P(B).
  • Bayes' theorem reverses a conditional: P(A | B) = P(B | A)P(A)/P(B), with P(B) found by the law of total probability.
  • When the thing being tested for is rare, most positive results can be false even from an accurate test; this is the base-rate effect.
  • Converting probabilities to counts out of 10,000 makes Bayes problems easier to set up and check.

Probabilities change when you learn something. Before two dice are rolled, a total of 10 has probability 3/36. Roll the first die and see a 5, and the chance of a 10 becomes 1/6. Conditional probability is the arithmetic of that update, and Bayes' theorem is the tool for running it in reverse. OpenStax's Contemporary Mathematics opens its section on the topic with exactly this dice example.

Conditional probability

The probability of A given B is

P(A | B) = P(A and B) / P(B), provided P(B) > 0.

The idea: once B is known to have happened, the outcomes outside B no longer matter. Restrict to the outcomes in B and ask what fraction of them are also in A.

Dice. Let A be "the total is 10" and B be "the first die shows 5". P(B) = 6/36. P(A and B) = 1/36, since only (5, 5) works. So P(A | B) = (1/36)/(6/36) = 1/6. If instead the first die shows 3, a total of 10 is impossible: P(A | first die 3) = 0.

Cards. Draw one card from a standard deck. The probability it is a king given it is a face card (jack, queen or king) is 4/12 = 1/3. The probability it is a face card given it is a king is 1. The two conditionals are different, so swapping P(A | B) for P(B | A) changes the answer completely.

The multiplication rule and independence

Rearranging the definition gives the multiplication rule:

P(A and B) = P(B) · P(A | B).

Drawing without replacement. Two cards from a deck: the probability both are aces is (4/52) · (3/51) = 12/2,652 = 1/221. The second factor is conditional, because one ace is gone.

Events are independent when knowing one tells you nothing about the other: P(A | B) = P(A), or equivalently P(A and B) = P(A) · P(B). Two separate dice are independent. Two cards drawn without replacement are not.

The law of total probability

If B₁, …, Bₖ split the possibilities into cases that cannot overlap, then

P(A) = P(A | B₁)P(B₁) + ⋯ + P(A | Bₖ)P(Bₖ).

In words: work out the chance of A in each case, weight by how likely each case is, and add. This is what makes Bayes' theorem usable.

Bayes' theorem

MathWorld describes Bayes' theorem as expressing a conditional probability in terms of its reverse and the corresponding marginal probabilities:

P(A | B) = P(B | A) · P(A) / P(B).

It follows in one line from the multiplication rule written both ways: P(A and B) = P(A | B)P(B) = P(B | A)P(A).

A worked test example

A factory makes chips, and 1% of them are faulty. An automated test flags 95% of faulty chips. It also wrongly flags 2% of good chips. A chip is flagged. What is the probability it is actually faulty?

Let F be "faulty" and T be "flagged". We know P(F) = 0.01, P(T | F) = 0.95 and P(T | not F) = 0.02.

Step 1: total probability of a flag.

P(T) = 0.95 · 0.01 + 0.02 · 0.99 = 0.0095 + 0.0198 = 0.0293.

Step 2: Bayes.

P(F | T) = 0.0095 / 0.0293 ≈ 0.324.

Only about a third of flagged chips are faulty, even though the test catches 95% of faults.

The same calculation as counts. Imagine 10,000 chips.

Flagged Not flagged Total
Faulty 95 5 100
Good 198 9,702 9,900
Total 293 9,707 10,000

Of the 293 flagged chips, 95 are faulty: 95/293 ≈ 32.4%. The table makes the reason visible. Faulty chips are rare, so the 2% false-alarm rate applied to 9,900 good chips produces more flags than the 95% detection rate applied to 100 faulty ones.

This is the base-rate effect: when the thing being looked for is rare, a positive result is weaker evidence than the test's accuracy suggests. The same arithmetic applies to spam filters, fraud alerts and any screening test.

What a second test does. If a flagged chip is retested independently and flagged again, run Bayes again with the new prior P(F) = 0.324: P(F | two flags) = (0.95 · 0.324)/(0.95 · 0.324 + 0.02 · 0.676) ≈ 0.958. Independent evidence accumulates: one flag left the chip probably good, two make it probably faulty.

A checklist for Bayes problems

  1. Name the events and write down every probability you are given, as P(something | something).
  2. Identify which conditional you want and which one you have; they are usually reversed.
  3. Find P(evidence) with the law of total probability.
  4. Apply Bayes.
  5. Sanity-check with counts out of 10,000.

The Monty Hall problem is a famous puzzle that this checklist solves cleanly. Conditional probabilities also feed into expected value calculations and into the binomial distribution, where independent trials multiply.

Probability is the eighth unit of Discrete Math: Study & Quiz, and its topic quizzes and review of missed questions can be used to practise problems like these. It sits with our other education apps, and the discrete math hub groups the probability guides together.

Good to know

Frequently asked questions

What is conditional probability?
The probability of an event given that another event has happened. It is computed by restricting the sample space to the outcomes where the given event occurred: P(A | B) = P(A and B)/P(B). Rolling two dice, the chance of a total of 10 is 3/36, but given the first die shows 5 it is 1/6.
What does Bayes' theorem say?
It turns P(B | A) into P(A | B): P(A | B) = P(B | A) · P(A)/P(B). It is used when you know how likely the evidence is under each explanation and want to know how likely each explanation is given the evidence.
Is P(A | B) the same as P(B | A)?
No, and they can be very different. The probability that a card is a king given it is a face card is 4/12, but the probability that a card is a face card given it is a king is 1. Bayes' theorem is the formula that connects the two.
What is the base rate fallacy?
Ignoring how common something is when interpreting evidence for it. If only 1% of items are faulty, even a test that flags 95% of faulty items and only 2% of good ones produces more false alarms than true ones, because good items vastly outnumber faulty ones.

Sources

  1. 7.9 Conditional Probability and the Multiplication Rule (Contemporary Mathematics) (opens in a new tab)OpenStax — accessed
  2. Bayes' Theorem (opens in a new tab)Wolfram MathWorld — accessed
  3. Conditional Probability (opens in a new tab)Wolfram MathWorld — accessed

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