Question

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

• A. $\frac{1}{3}$
• B. $\frac{1}{6}$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is A

$\frac{1}{3}$

Find the required probability

Only two tests are required when the two machines tested are either both working or both are not working.

Probability $\left(\mathrm{W}\right)$ when both are working,

$\begin{array}{rcl}\mathrm{W}& =& \frac{2}{4}\times \frac{1}{3}\\ \mathrm{W}& =& \frac{2}{12}\end{array}$

Probability $\left(\mathrm{F}\right)$ when both are not working

$\begin{array}{rcl}\mathrm{F}& =& \frac{2}{4}\times \frac{1}{3}\\ \mathrm{F}& =& \frac{2}{12}\end{array}$

Therefore total probability that only two test are required is

$$\begin{gathered} \frac{2}{12}+\frac{2}{12} \\ =\frac{4}{12} \\ =\frac{1}{3} \end{gathered}$$

Hence option $\left(\mathrm{A}\right)$, $\frac{1}{3}$ is the correct answer.