Answer to Question #199070 in Statistics and Probability for Ryan

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 $\bar{A}$ denotes machine I is inoperative and $\bar{B}$ denotes machine II is inoperative.

Given, $P(\bar{A})=1/3, P(A)=1-P(\bar{A})=1-1/3=2/3$

$P(\bar{B})=1/4, P(B)=1-P(\bar{B})=1-1/4=3/4$

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

$$P(\text{at least one will become inoperative })$$

$$=1-P(A\cap B)=1-P(A)P(B)$$

$$=1-\dfrac{2}{3}\times\dfrac{3}{4}=\dfrac{1}{2}$$