Conditional Probability

Unit 4 · Probability, Random Variables & Distributions

What AP Stats asks here

Conditional probability restricts attention to the world where $B$ has occurred. The recurring trap is asymmetry — $P(A \mid B)$ is not generally $P(B \mid A)$ — and the canonical example is medical testing where 'sensitivity' and 'positive predictive value' are easily confused. The companion question is the denominator: 'among those who...' means condition on that subgroup.

Conditional and multiplication

Conditional

P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}

Multiplication

P(A \cap B) = P(A) \cdot P(B \mid A)

Asymmetric

P(A \mid B) \neq P(B \mid A) \text{ in general}

AP Tip: Phrases like 'of those who...' or 'among...' signal that the denominator is a subgroup, not the population. Treat that phrase as a flashing 'condition on this'.

Caution: A 'positive on a 95% accurate test' does not mean 'the patient has the disease with probability 0.95'. The base rate matters: when the disease is rare, most positives are still false positives.

Type 1

Conditional from a two-way table

Identify the conditioning event, restrict attention to that row or column, then take the relevant cell over the row or column total.

Example 1

200 students: 100 studied (80 passed, 20 failed), 100 did not study (30 passed, 70 failed). What is

P(\text{passed} \mid \text{studied})

? (A)

80/200 = 0.40

(B)

80/110 \approx 0.73

(C)

80/100 = 0.80

(D)

30/100 = 0.30

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Type 2

Asymmetry: $P(A \mid B) \neq P(B \mid A)$

Direction matters. Compute one direction and invert via the multiplication rule.

Example 2

In a class, 40% are sophomores and 25% play sports. Among sophomores, 40% play sports. What is

P(\text{sophomore} \mid \text{plays sports})

? (A) 0.16 (B) 0.25 (C) 0.40 (D) 0.64

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Type 3

Multiplication rule and sequential draws

Sequential events without replacement are dependent. Use $P(A) \cdot P(B \mid A)$ to chain them.

Example 3

A box has 10 chocolates: 6 dark and 4 milk. Two are drawn without replacement. What is

P(\text{both dark})

? (A)

6/10 \times 6/10 = 0.36

(B)

6/10 \times 5/9 = 1/3 \approx 0.333

(C)

6/10 + 5/9 \approx 1.156

(D)

6/10 \times 4/9 \approx 0.267

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Conditional Probability

What AP Stats asks here

Conditional and multiplication

Conditional from a two-way table

Asymmetry: P(A∣B)≠P(B∣A)P(A \mid B) \neq P(B \mid A)P(A∣B)=P(B∣A)

Multiplication rule and sequential draws

Asymmetry: $P(A \mid B) \neq P(B \mid A)$