Question

There is a three-volume dictionary among 40 books arranged on a shelf in random order. The probability of these volumes standing in increasing order from left to right (the volumes are not necessarily kept side by side) is

• A. $\frac{1}{3}$
• B. $\frac{1}{6}$
• C. $\frac{1}{9}$
• D. none

Answer 1

The correct option is B $\frac{1}{6}$

Given total no. of books $n=40$

The $3$ volume dictionary $=m$

$\Rightarrow P\left(E\right)=\frac{m_1}{n}=\frac{1}{40}$ $P\left(F\right)=\frac{1}{40}$ $\therefore P(E\cap F)=\frac{3}{40}$

$\Rightarrow P\left(\frac{E}{F}\right)=\frac{3}{1}$ $P\left(\frac{E}{F}\right)=\frac{3}{1}$

$\Rightarrow P(E\cap F)=\frac{m}{n}=\frac{m}{m_1}\times \frac{m_1}{n}=\frac{3}{40}$

$\Rightarrow P(E\cap F)=\frac{m}{n}=\frac{m}{m_2}\times \frac{m_2}{n}=\frac{3}{40}$

$=\frac{3}{40}+\frac{3}{40}=\frac{6}{40}$

$\Rightarrow P(E\cap F)=P\left(E\right)\times P\left(\frac{F}{E}\right)=\frac{\left(\frac{1}{40}\right)}{\left(\frac{6}{40}\right)}$

$\therefore P\left(\frac{E}{F}\right)=\left(\frac{1}{6}\right).$

Hence, the answer is $\frac{1}{6}.$