Conditional Probability

Once you learn that something has already happened, the probability of the very same event can change. Conditional probability shrinks the sample space down to B and looks at A again under that condition, and from there the multiplication rule and Bayes' theorem follow naturally. Bayes in particular lets you reason backward from an effect to its cause, explaining why a positive test can still mean a low real chance when the disease is rare. Drag the P(A), P(B), and P(A∩B) sliders to see how P(A|B) shifts, then use the prevalence slider to uncover what a positive result truly means.

Conditions Change Probability

🔍 What is Conditional Probability?

①Probability of A given B has occurred

②Sample space shrinks from S to B

③Putting on B-tinted glasses to look at A again

Visualize P(A|B)

P(A)×104

P(B)×105

P(A∩B)×102

Definition

P(A|B) = P(A ∩ B)P(B)

Probability of A given B (with P(B) > 0)

Multiplication Rule

P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A)

Probability that both occur

Bayes Theorem

Disease prevalence (%)1%

🏥 Medical Test Paradox

①With 1% prevalence, a positive may still mean low actual probability of disease

②Positive test → truly sick? That is Bayes

③Low prior → low posterior

Bayes

P(A|B) = P(B|A) × P(A)P(B)

Infer cause probability from observed effect

Work It Out

Example 1

If P(A) = 1/2 and P(A∩B) = 1/3, find the conditional probability P(B|A).

Write the definition of conditional probability.

P(B|A) = P(A∩B)/P(A)

Substitute.

= (1/3)/(1/2) = 2/3

▸ 2/3

Conditional probability is the probability of B given that A has occurred.

Example 2

In a class of 30, 18 wear glasses and 12 of those are boys. Picking one glasses-wearer, find the probability of a boy.

Restrict the sample space to the condition (the 18 glasses-wearers).

P(boy | glasses) = (boys with glasses)/(glasses-wearers)

Substitute.

= 12/18 = 2/3

▸ 2/3

Conditional probability narrows the sample space to the given condition.

Wrap-up

Core

P(A|B) = P(A ∩ B)P(B), P(A|B) = P(B|A)P(A)P(B)

Conditional probability + Bayes

Education-office assessment type

For events A, B with P(A) = 0.4 and P(B|A) = 0.5, find P(A∩B).

①0.1

②0.2

③0.3

④0.4

⑤0.5

▸ ② 0.2

Use the multiplication rule of probability.

P(A∩B) = P(A) × P(B|A)

Substitute.

= 0.4 × 0.5 = 0.2

🎯 Exam Points

①Conditional: P(A|B) = P(A∩B)/P(B)

②Multiplication: P(A∩B) = P(B)·P(A|B)

③Independent ⇒ P(A|B) = P(A)

④Bayes: outcome → cause inference

⑤Tree diagrams decompose total probability

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Meaning of Probability

Independent Trials

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