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MathidiaMathematics Academy
Probability
Lesson 37 of 75
Easy practice

Conditional probability

Objective. Use conditioning, partitions and Bayes' theorem.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

Conditional probability measures how the likelihood of an event changes given that another event is known to have occurred, defined as P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. A partition of the sample space into mutually exclusive, exhaustive events Bᵢ allows the law of total probability: P(A) = Σ P(A|Bᵢ)P(Bᵢ), summing contributions across every scenario. Bayes' theorem then reverses the conditioning: P(B|A) = P(A|B)P(B)/P(A), letting you update beliefs about B after observing A. For a disease test with sensitivity 0.9 and a rare disease with prevalence 0.01, Bayes' theorem reveals that a positive test result still carries a surprisingly low true-positive probability once the rarity is properly accounted for.

Worked example

A bag has 5 red and 3 blue balls. Given two balls are drawn without replacement, find the conditional probability the second is red given the first was red.

5 … 3
Illustrated card showing 5 … 3
  1. 1After drawing one red ball, 4 red and 3 blue balls remain, totalling 7 balls.
  2. 2The conditional probability only concerns the second draw given this reduced bag.
  3. 3Count favourable outcomes: 4 red balls out of 7 total remaining balls.
  4. 4Write the conditional probability as the ratio of these counts.

Answer: 4/7

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Common mistake. Students frequently confuse P(A|B) with P(B|A), treating conditional probability as symmetric; the probability of rain given clouds is very different from the probability of clouds given rain.

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