Questions: Independence and the Multiplication Rule
5 questions to test your understanding
Score: 0 / 5
Question 1 Multiple Choice
A student draws one card from a shuffled 52-card deck, notes it is a king, and does not replace it. What is the probability that the second card drawn is also a king?
A4/52, because each draw is an independent event
B3/51, because knowledge of the first draw changes the probability for the second
C1/13, because kings always appear at a fixed rate
D0, because two kings cannot be drawn in sequence
Because the card is not replaced, the two draws are dependent: knowing the first card was a king changes the composition of the remaining deck (51 cards, 3 kings). So P(2nd king | 1st king) = 3/51 ≠ 4/52. Option A is the misconception — treating draws without replacement as independent. Independence requires that P(A|B) = P(A), which fails here.
Question 2 Multiple Choice
Events A and B satisfy P(A ∩ B) = 0, and both P(A) > 0 and P(B) > 0. Which of the following must be true?
AA and B are independent
BA and B are mutually exclusive and therefore not independent
CA and B are independent because they share no outcomes
DA and B are both equally probable
P(A ∩ B) = 0 means A and B are mutually exclusive — they cannot both occur. But this makes them maximally dependent, not independent: if A occurs, you know with certainty that B did not (P(B|A) = 0 ≠ P(B)). Mutual exclusivity and independence are essentially opposite concepts. Independence requires that P(A|B) = P(A), which is violated whenever two positive-probability events are mutually exclusive.
Question 3 True / False
If two events are independent, they cannot both occur at the same time.
TTrue
FFalse
Answer: False
That describes mutual exclusivity, not independence. Independent events can absolutely co-occur — flipping heads on a coin and rolling an even number on a die are independent, and both happen 25% of the time. Independence means P(A|B) = P(A): knowing one event occurred gives no information about the other. Mutual exclusivity means P(A ∩ B) = 0: knowing one occurred tells you the other definitely did not.
Question 4 True / False
Two mutually exclusive events with positive probability are independent.
TTrue
FFalse
Answer: False
If A and B are mutually exclusive, P(A ∩ B) = 0. But P(A) · P(B) > 0 since both events have positive probability. Therefore P(A ∩ B) ≠ P(A) · P(B), which means A and B are not independent. Equivalently, P(B|A) = 0 ≠ P(B), so knowing A occurred tells you B definitely did not — the opposite of independence.
Question 5 Short Answer
Why does the formula P(A ∩ B) = P(A) · P(B) only work when A and B are independent, and what formula applies in the general case?
Think about your answer, then reveal below.
Model answer: P(A) · P(B) is derived from the definition of independence: P(A|B) = P(A). Substituting into the definition of conditional probability gives P(A ∩ B) = P(A) · P(B). When events are dependent, P(A|B) ≠ P(A), so this simplification is invalid. The general multiplication rule — P(A ∩ B) = P(A) · P(B|A) — applies whether or not events are independent; when they are independent, P(B|A) = P(B) and it reduces to the product formula.
The product rule is a special case, not the general rule. The mistake of applying P(A) · P(B) to dependent events (like card draws without replacement) gives wrong answers because it ignores how the occurrence of A changes the probability of B. The general multiplication rule forces you to account for that dependency explicitly.