Questions: Independence and Mutually Exclusive Events
5 questions to test your understanding
Score: 0 / 5
Question 1 Multiple Choice
Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.4. Are A and B independent?
AYes — their Venn diagram circles don't overlap, so they have nothing to do with each other
BYes — P(A ∩ B) = 0, which satisfies the independence condition
CNo — knowing A occurred makes B impossible, so they are maximally dependent
DIt depends on whether P(A ∪ B) = 1
Mutually exclusive events with nonzero probability are always dependent. Independence requires P(A|B) = P(A). But if A and B are mutually exclusive, P(A|B) = P(A ∩ B)/P(B) = 0/0.4 = 0, which does not equal P(A) = 0.3. Knowing B occurred completely rules out A — that is the opposite of independence. The intuitive confusion arises from equating 'don't overlap on a Venn diagram' with 'have nothing to do with each other,' but probabilistic independence is not about overlap — it is about information.
Question 2 Multiple Choice
Which formula correctly tests whether events A and B are independent?
AP(A ∩ B) = 0
BP(A ∪ B) = P(A) + P(B)
CP(A ∩ B) = P(A) · P(B)
DP(A|B) = P(B|A)
Independence is defined as P(A|B) = P(A), which rearranges to P(A ∩ B) = P(A) · P(B) — the product rule. Options A and B describe mutually exclusive events (where P(A ∩ B) = 0 and the addition rule has no subtraction term), not independence. Option D describes symmetry of conditional probability, which holds in general and does not characterize independence.
Question 3 True / False
Mutually exclusive events are independent because, since they cannot occur simultaneously, neither one can influence the other.
TTrue
FFalse
Answer: False
This is the central misconception. Mutual exclusivity makes events maximally dependent, not independent. If A occurs, B is ruled out entirely — that is the strongest possible information one event can provide about another. Independence means observing one event gives zero information about the other, which requires P(A ∩ B) = P(A)·P(B) > 0. Mutual exclusivity forces P(A ∩ B) = 0, which violates independence whenever both events have positive probability.
Question 4 True / False
If P(A) = 0.5 and P(A|B) = 0.5, then A and B are independent, regardless of whether their Venn diagram circles overlap.
TTrue
FFalse
Answer: True
Independence is fully defined by the condition P(A|B) = P(A). If this condition holds, then observing B provides no information about A — that is exactly independence. The Venn diagram overlap (whether P(A ∩ B) > 0) is irrelevant to the independence condition. In fact, if P(A|B) = P(A) = 0.5 and P(B) > 0, then P(A ∩ B) = P(A|B)·P(B) = 0.5·P(B) = P(A)·P(B), confirming the product rule.
Question 5 Short Answer
Explain why two mutually exclusive events, each with nonzero probability, must be dependent. Use the definition of independence in your explanation.
Think about your answer, then reveal below.
Model answer: Independence requires P(A|B) = P(A). For mutually exclusive events, P(A ∩ B) = 0, so P(A|B) = P(A ∩ B)/P(B) = 0/P(B) = 0. But P(A) > 0 by assumption, so P(A|B) = 0 ≠ P(A). The condition for independence fails. In fact, knowing B occurred drops the probability of A from P(A) to 0 — the maximum possible change — making them maximally dependent.
The key is applying the definition precisely. Independence is not a spatial or visual concept (about diagram overlap) but an informational one: does knowledge of B change your probability for A? For mutually exclusive events, it changes it as much as possible — all the way to 0. The confusion between 'disjoint' and 'unrelated' is the source of the misconception.