Problem 7 · 2015 AMC 8
Easy
Counting & Probability
complementary-counting
Each of two boxes contains three chips numbered 1, 2, 3. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
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Answer: E — 5/9.
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Hint 1 of 2
'Even product' happens in lots of ways (either chip even is enough), but 'odd product' happens in just one tidy way: both chips must be odd. Chase the easy case.
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Hint 2 of 2
Compute the easy event and subtract from 1: this is complementary counting. Each box has 2 odd values out of 3, and the boxes are independent, so multiply.
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Approach: complementary counting — find P(odd) and subtract
- A product is odd only when both factors are odd, while it's even in many scenarios — so the odd case is the simpler one to count.
- Each box has odds {1, 3} out of {1, 2, 3}, a 2/3 chance, and the draws are independent: P(both odd) = (2/3)(2/3) = 4/9.
- P(even product) = 1 − 4/9 = 5/9.
- Why this transfers: 'at least one __' or 'is even/divisible' events are usually faster through the complement — count the one clean way it fails, then subtract from 1.
Another way — direct count over all 9 outcomes:
- The two draws give 3 × 3 = 9 equally likely pairs.
- List products: row by row they are 1,2,3 / 2,4,6 / 3,6,9. The even ones are 2,2,4,6,6 — that's 5 outcomes.
- Probability = 5/9 = 5/9, matching the complement.
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