First slide

Random variable and probability distribution

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 ${(\frac{3}{4})}^{3} k$ then k is equal to:

Moderate

A

$\frac{17}{2}$
B

4
C

$\frac{17}{8}$
D

$\frac{17}{4}$

Solution

Required probability = when no. machine has fault + when only one machine has fault + when only two machines have fault.
$=^{5} C_{0} {(\frac{3}{4})}^{5} +^{5} C_{1} (\frac{1}{4}) {(\frac{3}{4})}^{4} +^{5} C_{2} {(\frac{1}{4})}^{2} {(\frac{3}{4})}^{3}$
$= \frac{243}{1024} + \frac{405}{1024} + \frac{270}{1024} = \frac{918}{1024} = \frac{459}{512} = \frac{27 \times 17}{64 \times 8}$
$= {(\frac{3}{4})}^{3} \times k = {(\frac{3}{4})}^{3} \times \frac{17}{8}$
$∴ k = \frac{17}{8}$

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