Independence

When Events Don't Influence Each Other

Events $A$ and $B$ are independent if knowing $B$ occurred tells you nothing about $A$:

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

Equivalently (and more useful in practice):

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

# Two fair coin flips are independent
p_heads_1 = 0.5
p_heads_2 = 0.5
p_both = p_heads_1 * p_heads_2

print(p_both)  # 0.25 — P(HH)

Testing for Independence

Given $P(A)$, $P(B)$, and $P(A \cap B)$, events are independent iff:

$|P(A \cap B) - P(A) \cdot P(B)| < \epsilon$

Mutual Exclusivity ≠ Independence

Mutually exclusive events (where $P(A \cap B) = 0$) are almost never independent (unless one has probability 0). If $A$ occurs, $B$ definitely cannot — that is dependence.

Your Task

Implement two functions:

1. are_independent(p_a, p_b, p_ab) — returns True if $|P(A \cap B) - P(A) \cdot P(B)| < 10^{-9}$
2. joint_independent(p_a, p_b) — returns $P(A) \cdot P(B)$, rounded to 4 decimal places

Print the independence check, then the joint probability.