Q.421. Probability.

Question 421.
A 4" * 4" * 4" cube is painted and then cut into 64 1" * 1" * 1" cubes. A unit cube is then randomly selected and rolled. What is the probability that the top face of the rolled cube is painted. Give answer as a common fraction.

Answer 421.
There are 8 cubes with three faces painted, 24 with two faces pained, 24 with one face painted and 8 with no faces pained.

Let A = event that a cube with three colored faces is selected and rolled
B = event that a cube with two colored faces is selected and rolled
C = event that a cube with one colored faces is selected and rolled and
D = event that a cube with no colored face is selected and rolled
E = event that the randomly selected and rolled cube has top face pained
=> P(A) = 8/64, P(B) = 24/64, P(C) = 24/64 and P(D) = 8/64
and P(E/A) = 1/2, P(E/B) = 1/3, P(E/C) = 1/6 and P(E/D) = 0
=> required probability
= (1/2) * (8/64) + (1/3) * (24/64) + (1/6) * (24/64)
= 4/64 + 8/64 + 4/64
= 1/4.

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