Question

In a book of $500$ pages, it is found that there are $250$ typing errors.

Assume that Poisson law holds for the number of errors per page.

Then, the probability that a random sample of $2$ pages will contain no error, is

• A. $e^{-0.3}$
• B. $e^{-0.5}$
• C. $e^{-1}$
• D. $e^{-2}$

Answer 1

The correct option is C

$e^{-1}$

Explanation for the correct option.

Find the probability.

In a book containing $500$ pages, there are $250$ typing errors

Now, $\lambda$ is defined as total pages in the book by number of errors.

$\begin{array}{rcl}\lambda & =& \frac{500}{250}\\ & =& 2\end{array}$

Let the sample size be $n$ and $r$ be the number of errors.

Now, the probability that exactly $r$ error occurs in a sample size $n$ pages is given as: $P(x=r)=\frac{e^{-{\displaystyle \frac{\lambda }{n}}}{\lambda }^r}{r!}$.

For the question, the sample size is $2$ and number of errors is $0$.

So, substitute $r=0$ and $n=2$.

$\begin{array}{rcl}P(x& =& 0)=\frac{e^{-{\displaystyle \frac{\lambda }{2}}}{\lambda }^0}{0!}\\ & =& \frac{e^{-{\displaystyle \frac{2}{2}}}\times 1}{1}\left[\lambda =2\right]\\ & =& e^{-1}\end{array}$

So, there is a probability of $e^{-1}$ that there will be no error in a random sample of $2$ pages.

Hence, the correct option is C.