In a workshop there are five machines and the probability of any one of them to be out of service on a day is . If the probability that at most two machines will be out of service on the same day is , is equal to

JEE MainMathematicsModerateProbability

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 $\left( \frac{3}{4} \right)^{3}k$ , is equal to

Show answer and solution

Required probability = when no. machine has fault + when only
one machine has fault + when only two machines have fault.

= \ ^{5}C_{0}\left( \frac{3}{4} \right)^{5} + \ ^{5}C_{1}\left( \frac{1}{4} \right)\left( \frac{3}{4} \right)^{4} + \ ^{5}C_{2}\left( \frac{1}{4} \right)^{2}\left( \frac{3}{4} \right)^{3}

= \frac{243}{1024} + \frac{405}{1024} + \frac{270}{1024} = \frac{918}{1024} = \frac{459}{512} = \frac{27 \times 17}{64 \times 8}

= \left( \frac{3}{4} \right)^{3} \times k = \left( \frac{3}{4} \right)^{3} \times \frac{17}{8}

\therefore k = \frac{17}{8}

Source: Grass-G / M-08 Mathematics Paper 23 October.docx

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