Problem 12 · 2014 AMC 8
Easy
Counting & Probability
permutationsprobability-basic
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
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Answer: B — 1/6.
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Hint 1 of 2
There's exactly one correct matching, so the probability is just 1 ÷ (number of possible matchings). The whole problem is counting the orderings.
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Hint 2 of 2
Three baby photos can be lined up against the three celebrities in 3! ways.
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Approach: one favorable outcome over all equally likely orderings
- Order the 3 baby photos against the celebrities: 3! = 6 equally likely ways.
- Exactly 1 of those 6 is the all-correct matching, so probability = 1/6 = 1/6.
- Reusable idea: when every arrangement is equally likely and only one wins, P(win) = 1/(total arrangements). The hard part is always the count, never the division.
Another way — match one celebrity at a time:
- First celebrity: 1 of 3 baby photos is right ⇒ chance 1/3.
- Given that, the second celebrity: 1 of the 2 remaining is right ⇒ chance 1/2 (and the third is then forced).
- Multiply: (1/3)(1/2) = 1/6.
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