Question

A candiate is selected for interview for three posts.For the first there are 3 candidates, for the second there are 4 and for the third there are 2. What are the chances of his getting at least one post ?

• A. $\frac{3}{4}.$
• B. $\frac{11}{12}.$
• C. $\frac{1}{4}.$
• D. $\frac{7}{8}.$

Answer 1

Answer: (A)

$\frac{3}{4}.$

Let A, B and C denote the events that the candidate gets the first , second and third post respectively.
$\therefore P\left(A\right)=\frac{1}{3}$
$\Rightarrow P\left(\overline{A}\right)=1-P\left(A\right)=\frac{2}{3}$
$P\left(B\right)=\frac{1}{4}$
$\Rightarrow P\left(\overline{B}\right)=\frac{3}{4}$
$P\left(C\right)=\frac{1}{2}$
$\Rightarrow P\left(\overline{C}\right)=\frac{1}{2}$
Now , probability that candidate will get at least one post is $P(A\cup B\cup C)$
$=1-P\left(\overline{A\cup B\cup C}\right)$
$=1-P(\overline{A}\cap \overline{B}\cap \overline{C})$
$=1-P\left(\overline{A}\right)P\left(\overline{B}\right)P\left(\overline{C}\right)$ (Since events A, B, C are independent $\Rightarrow \overline{A},\overline{B},\overline{C},$ are also independent)
$=1-\frac{2}{3}\times \frac{3}{4}\times \frac{1}{2}=\frac{3}{4}.$