Question

Suppose that two machines I and II in a factory operate independently of each other. Past

experience showed that during a given 8-hour time, machine I remains inoperative one

third of the time and machine II does so about one fourth of the time. What is the

probability that at least one of the machines will become inoperative during the given

period?

Answer 1

Let $I$ denotes machine I is inoperative and $II$ denotes machine II is inoperative.

Given, $P(I)=1/3, P(I')=2/3$

$P(II)=1/4, P(II')=3/4$

Now, the probability that at least one of the machines will become inoperative during the given period$=P(I)P(II')+P(I')P(II)+P(I)P(II)$

$=\dfrac{1\times3}{3\times4}+\dfrac{2\times1}{3\times4}+\dfrac{1\times1}{3\times4}$

$=\dfrac{1}{2}$