2018 AMC 10B Problem 9

The faces of each of

7

standard dice are labeled with the integers from

1

6.

Let

p

be the probability that when all

7

dice are rolled, the sum of the numbers on the top faces is

10.

What other sum occurs with the same probability

p?

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Solution

Replace each die’s value

k

7 - k.

This pairs up outcomes one-to-one and keeps their probabilities, and it sends a total of

s

7 \cdot 7 - s = 49 - s.

So the sums

s

and

49 - s

are equally likely. The partner of

10

49 - 10 = 39.

Thus, D is the correct answer.

Tagged: dice (probability) · bijection · symmetry