Events A and B are mutually exclusive, each with P(A) = P(B) = 0.3. Are A and B independent?
AYes — mutually exclusive events cannot influence each other since they share no outcomes
BNo — if B occurred, then A definitely did not occur, so P(A|B) = 0 ≠ 0.3 = P(A)
CYes — independence and mutual exclusivity are equivalent for events with equal probabilities
DIt depends on whether A and B are from the same experiment
Mutually exclusive events (A ∩ B = ∅) with positive probability are dependent, not independent. If B occurred, A is impossible — P(A|B) = 0, which differs from P(A) = 0.3. Independence requires that knowing B occurred gives no information about A. Mutual exclusivity gives maximum negative information: knowing B makes A impossible. This is the opposite of independence. The confusion arises because 'they can't happen together' sounds like 'they don't affect each other,' but probabilistically it's a strong dependency.
Question 2 Multiple Choice
Which of the following is the correct working definition of independence for events A and B?
AA and B come from physically separate experiments
BA and B cannot occur at the same time
CP(A ∩ B) = P(A) · P(B)
DP(A | B) is defined and equals P(B | A)
Independence is formally defined as P(A ∩ B) = P(A) · P(B). This product rule is the working definition because it avoids dividing by P(B) (which could be 0) and generalizes naturally to more than two events. Option A describes a common intuition that is often but not always correct — physical separateness suggests independence but is not the definition. Option B is mutual exclusivity, which implies dependence for events with positive probability.
Question 3 True / False
If P(A) = 0.4 and P(B) = 0.5 and A and B are independent, then P(A ∩ B) = 0.2.
TTrue
FFalse
Answer: True
By the product rule for independent events: P(A ∩ B) = P(A) · P(B) = 0.4 × 0.5 = 0.2. This is direct application of the definition. Note that this product rule is not valid for dependent events — if A and B were dependent, you would need P(A ∩ B) = P(A) · P(B | A), and P(B | A) ≠ P(B).
Question 4 True / False
If three events are pairwise independent — most pair satisfies the product rule — then the three events are mutually independent.
TTrue
FFalse
Answer: False
Pairwise independence does not imply mutual independence. Classic counterexample: flip a fair coin twice. Let A₁ = heads on flip 1, A₂ = heads on flip 2, A₃ = exactly one head total. Every pair is independent (P(Aᵢ ∩ Aⱼ) = P(Aᵢ)P(Aⱼ) for each pair). But P(A₁ ∩ A₂ ∩ A₃) = 0 (you can't have two heads AND exactly one head), while P(A₁)P(A₂)P(A₃) = 1/8 ≠ 0. Mutual independence requires all subsets — not just pairs — to satisfy the product rule.
Question 5 Short Answer
Explain why mutually exclusive events with positive probability are dependent, not independent. Use the formal definition of independence in your answer.
Think about your answer, then reveal below.
Model answer: Two events are independent if P(A ∩ B) = P(A) · P(B). For mutually exclusive events, P(A ∩ B) = 0 (they cannot both occur). But if P(A) > 0 and P(B) > 0, then P(A) · P(B) > 0. So P(A ∩ B) = 0 ≠ P(A) · P(B), violating the product rule — the events are dependent. Intuitively: if A and B are mutually exclusive and you learn B occurred, you immediately know A did not occur (P(A|B) = 0), which differs from P(A). Learning B tells you something definitive about A — the opposite of independence.
This is one of the most important and counterintuitive facts in probability. Students often confuse 'independent' with 'unrelated' or 'separate,' and 'mutually exclusive' sounds like the events are completely separate. But probability independence is a quantitative condition about information, and mutually exclusive events are as far from independent as possible.