Problem 13 · 2016 AMC 8
Easy
Counting & Probability
careful-counting
Two different numbers are randomly selected from the set {−2, −1, 0, 3, 4, 5} and multiplied together. What is the probability that the product is 0?
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Answer: D — 1/3.
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Hint 1 of 2
A product is 0 ONLY when one of its factors is 0. The minus signs and the size of the numbers are pure distraction — the only chip that matters is the 0. So really the question is: how often does 0 get picked?
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Hint 2 of 2
Count the pairs that include 0 versus all pairs. There are 6 numbers; pairing 0 with each of the other 5 gives the favorable pairs.
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Approach: a product is zero only when 0 is one of the two picks
- The product is 0 exactly when 0 is one of the two chosen numbers — the negatives and the spread of values don't matter at all.
- Total ways to choose 2 of the 6 numbers: C(6, 2) = 15. Pairs that contain 0: pair 0 with each of the other 5 numbers → 5 pairs.
- Probability = 5 / 15 = 1/3.
- Why this transfers: a "product equals 0" question always reduces to "is the special factor 0 chosen?" — ignore everything else and just count how often 0 appears.
Another way — probability the FIRST pick already settles it:
- Think of drawing one number, then a second. The chance 0 is NOT picked first is 5/6; given that, the chance 0 is not picked second is 4/5.
- So neither is 0 with probability (5/6)(4/5) = 4/6 = 2/3. The product is 0 the rest of the time: 1 − 2/3 = 1/3.
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