In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $ \frac{1}{4} $. If the probability that at most two machines will be out of service on the same day is $ \left(\frac{3}{4}\right)^{3} k $, then $ k $ is equal to

Question

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $ \frac{1}{4} $. If the probability that at most two machines will be out of service on the same day is $ \left(\frac{3}{4}\right)^{3} k $, then $ k $ is equal to, $ \frac{17}{8} $, $ \frac{17}{4} $, $ \frac{\sqrt{\frac{1}{2}}}{2} $, $ 4 $

MARKS App · Accepted Answer

Required probability = when no machine has fault + when one machine has fault + when two machines have fault. = C 0 5 3 4 5 + C 1 5 1 4 3 4 4 + C 2 5 1 4 2 3 4 3 = 243 1024 + 405 1024 + 270 1024 = 918 1024 = 459 512 = 27 × 17 64 × 8 ⇒ 3 4 3 × k = 3 4 3 × 17 8 ∴ k = 17 8

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