Problem 21 · 2004 AMC 8
Easy
Counting & Probability
complementary-counting
Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?
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Answer: D — 2/3.
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Hint 1 of 2
'Even product' can happen lots of ways (this even, that even, both even) — messy to count. But the opposite is razor-simple: a product is odd only when both spinners are odd. Count that one easy case and subtract.
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Hint 2 of 2
The technique is complementary counting: P(even) = 1 − P(odd), and a product is odd exactly when every factor is odd. Whenever 'at least one even' shows up, flip to 'all odd'.
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Approach: complement (1 minus all-odd)
- Spinner A shows 1, 2, 4, 3 — odds are {1, 3}, so P(A odd) = 2/4 = 1/2. Spinner B shows 1, 2, 3 — odds are {1, 3}, so P(B odd) = 2/3.
- Product is odd only if both are odd: P(odd) = (1/2)(2/3) = 1/3.
- Therefore P(even) = 1 − 1/3 = 2/3.
- Why the complement wins: the direct count needs three even-cases, but 'both odd' is a single product — one multiplication instead of several additions. Odd × odd = odd is the only way to dodge an even factor.
Another way — count favorable outcomes directly:
- There are 4 × 3 = 12 equally likely (A, B) outcomes.
- Odd products come only from A ∈ {1,3} and B ∈ {1,3}: that's 2 × 2 = 4 odd outcomes, so 12 − 4 = 8 even outcomes.
- P(even) = 8/12 = 2/3.
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