2018 AMC 10B Problem 9

Question

The faces of each of

7

standard dice are labeled with the integers from

1

to

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?

13

26

32

39

42

Replacing each die value

k

by

7 - k

pairs up equally likely outcomes

This replacement sends a total of

s

to a total of

49 - s

Replace each die’s value

k

by

7 - k.

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

s

to

7 \cdot 7 - s = 49 - s.

So the sums

s

and

49 - s

are equally likely. The partner of

10

is

49 - 10 = 39.

Thus, D is the correct answer.

Problem 9 in other years