Conditional probability: $P(A|B)$ without getting it wrong

The formula

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} \quad \text{(with } P(B) > 0\text{)}$

In English: among the cases where B happens, what fraction also realizes A.

The mental trick: « restrict the universe »

When you see $P(A \mid B)$, imagine that you ignore everything that isn’t B. Then count the proportion of A in this new universe.

A concrete example

In a class of 30:

• 12 are girls.
• 8 students wear glasses, including 5 girls.

$P(\text{glasses} \mid \text{girl}) = ?$

Restrict to girls: 12 students. Among them, 5 wear glasses. So $P = 5/12$.

Verify with the formula

• $P(\text{glasses} \cap \text{girl}) = 5/30$.
• $P(\text{girl}) = 12/30$.
• $P(\text{glasses} \mid \text{girl}) = \dfrac{5/30}{12/30} = 5/12$ ✓.

The costly mistake

$P(A \mid B) \ne P(B \mid A)$ in general.

Example: $P(\text{being black} \mid \text{being a raven})$ ≈ 100%. But $P(\text{being a raven} \mid \text{being black})$ is tiny.

Bayes’ formula (bonus)

$P(A \mid B) = \dfrac{P(B \mid A) \cdot P(A)}{P(B)}$

Key of medicine, machine learning, and decision theory.

?Your turn

You draw a card from a deck of 52. Given the card is red, what's the probability it's a heart?

A.1/4B.1/2C.1/13D.13/52

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