2009 AMC 10B Problem 25

Question

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?

\dfrac18

\dfrac{3}{16}

\dfrac14

\dfrac38

\dfrac12

Answer: B

Concepts:basic probability independent events symmetry

Difficulty rating: 2090

Small hint:

Each face has

2

equally likely stripe orientations, so

2^6

total configurations

Big hint:

There are

3

possible encircling bands, each fixing the orientation of

4

faces

Solution:

Each face’s stripe has

2

orientations, giving

2^6=64

equally likely configurations.

An encircling stripe runs around one of the

3

pairs of opposite faces. For a given band, the

4

faces it crosses must each be oriented to continue it, a probability of

\left(\dfrac12\right)^4=\dfrac{1}{16}.

The three bands are mutually exclusive, so the probability is

3\cdot\dfrac{1}{16}=\dfrac{3}{16}.

Thus, the correct answer is B.