Conditional Probability

Updating on Evidence

Conditional probability $P(A \mid B)$ is the probability of $A$ given that $B$ has occurred:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

Conditioning on $B$ reduces the sample space to only outcomes where $B$ holds.

# Roll two dice. Given the sum > 7, what is P(first die = 6)?
# Outcomes where sum > 7: (2,6),(3,5),(3,6),(4,4),(4,5),(4,6),(5,3)...(6,6) → 15 outcomes
# Of these, first die = 6: (6,2),(6,3),(6,4),(6,5),(6,6) → 5 outcomes
# P(first=6 | sum>7) = 5/15 = 1/3

p_ab = 5/36   # P(first=6 AND sum>7)
p_b  = 15/36  # P(sum > 7)
print(round(p_ab / p_b, 4))  # 0.3333

Chain Rule

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

This extends to any number of events: $P(A_1 \cap A_2 \cap \cdots \cap A_n) = P(A_1) \cdot P(A_2 \mid A_1) \cdot P(A_3 \mid A_1, A_2) \cdots$

Your Task

Implement conditional(p_ab, p_b) that returns $P(A \mid B) = P(A \cap B) / P(B)$, rounded to 4 decimal places.