Cubes

Probability Level pending

The exterior of a cube of side length 3 cm is painted red, and is then divided into 27 unit cubes. If one of these cubes is randomly selected and then rolled, what is the probability that a red face comes up?

2 3 \frac{2}{3} 1 3 \frac{1}{3} 1 2 \frac{1}{2} 1 4 \frac{1}{4}

Benedict Dimacutac by Benedict Dimacutac , 21, Philippines

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1 solution

Rohit Ner
Jan 20, 2016

On dividing the cube in 27 identical units, the cubes obtained can be classified as
8 8 cubes with 3 3 red faces,
12 12 cubes with 2 2 red faces,
6 6 cubes with 1 1 red face,
1 1 cube with 0 0 red faces.
P ( E ) = ( 8 27 ) ( 3 6 ) + ( 12 27 ) ( 2 6 ) + ( 6 27 ) ( 1 6 ) + ( 1 27 ) ( 0 6 ) = 1 3 \begin{aligned}P(E)&=\left(\frac{8}{27}\right)\left(\frac{3}{6}\right)+\left(\frac{12}{27}\right)\left(\frac{2}{6}\right)+\left(\frac{6}{27}\right)\left(\frac{1}{6}\right)+\left(\frac{1}{27}\right)\left(\frac{0}{6}\right)\\&\huge\color{#3D99F6}{=\boxed{\frac{1}{3}}}\end{aligned}

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