A 3x3x3 wooden cube is painted of all six faces, then cut into 27 unit cubes. One unit cube is randomly selected and rolled. After it is rolled, 5 of the 6 faces of the cube are visible, with the face on the bottom unseen. What is the probability that exactly one of the five faces is painted.

Guest Jun 26, 2018

#1**+3 **

**We first need to determine how the 27 unit cubes are painted. **

Number of Painted Sides | Number of Cubes |

3 | The 8 corner cubes |

2 | 12 non-corner edge cubes |

1 | 6 visible center cubes |

0 | 1 non-visible center cube |

**Cases:**

Case 1: We select a cube with 0 painted sides

It is not possible to see a painted side, since none are painted.

Case 2: We select a cube with 1 painted sides

The roll is successful as long as an unpainted side is on the bottom.

Case 3: We select a cube with 2 painted sides

The roll is successful as long as a painted side is on the bottom.

Case 4: We select a cube with 3 painted sides

It is not possible to see only one painted side, since only on can be hidden, but the other two cannot. Therefore, we will see more than one painted side.

**Probabilty:**

We only need to consider cases 1 and 2.

Case 2:

6 sides to a cube, 5 unpainted and 1 painted.

We need an unpainted side on the bottom, 5/6.

Case 3:

We need a painted side of the bottom, 2/6.

\(\frac{12}{27}\cdot\frac26+\frac{6}{27}\cdot\frac56=\boxed{\frac13}\)

I hope this helped,

Gavin

GYanggg
Jun 26, 2018

#2**+2 **

Thanks, Gavin....!!!!

Here's another way to see this....

Note that there are 9 faces on each side of the large cube that are painted....so....the total number of painted faces is just 9 * 6 = 54 painted faces

And the total number of faces on the smaller cubes is just 27 cubes * 6 faces on each = 162

If we put all the cubes in a bag and selected only one, the probability that it would have a painted face is just 54 /162 = 1 / 3

CPhill
Jun 26, 2018

#3**+1 **

Hey CPhill!

I don't think your answer accounts for this part of the question:

*After it is rolled, 5 of the 6 faces of the cube are visible, with the face on the bottom unseen. What is the probability that exactly one of the five faces is painted.*

In your case there isn't the rolling process, hence the "*5 of the 6 faces of the cube are visible, with the face on the bottom unseen" *part was neglected.

I'm still trying to find out why you also got the same answer.

I might have missed something about your solution and I'm the one at fault, and you solution is actually correct.

GYanggg
Jun 26, 2018