2009 AMC 10A Problem 22

Question

Two cubical dice each have removable numbers

1

through

6.

The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is

7?

\dfrac{1}{9}

\dfrac{1}{8}

\dfrac{1}{6}

\dfrac{2}{11}

\dfrac{1}{5}

Answer: D

Concepts:dice (probability)basic probability symmetry

Difficulty rating: 1820

Small hint:

The whole process is equivalent to picking two of the twelve tiles at random and adding them

Big hint:

After the first tile shows

N,

count how many of the remaining

11

tiles equal

7 - N

Solution:

Randomly attaching the tiles and then rolling is equivalent to choosing two of the twelve numbers at random and adding them.

Suppose the first top face shows

N.

For a sum of

7,

the second must be

7 - N,

and there are exactly

2

tiles equal to

7 - N

among the remaining

11.

So the probability is

\dfrac{2}{11}.

Thus, the correct answer is D.