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

Hint: Here in this question, it is given itself that it is a question of probability and we have to solve this by Poisson’s method. We have a particular formula to solve this type of question, we just have to put the values of different variables in the formula and then we get our answer.

Formula used:
$P\left( {X = r} \right) = \dfrac{{{e^{ - \dfrac{\lambda }{n}}}{\lambda ^r}}}{{r!}}$
Where, $r = \text{number of errors}$, $\lambda \, = \,\dfrac{\text{Total number of pages}}{\text{error}}$ and $n\, = \text{samples}$.


Complete step by step answer:
In the given question, we have to find the probability for a sample of $2$ pages that contains no error. So, we will use a formula to find the probability, that is,
$P\left( {X = r} \right) = \dfrac{{{e^{ - \dfrac{\lambda }{n}}}{\lambda ^r}}}{{r!}}$
From given question, we know that
$\text{number of errors} = 0\,\text{Total number of pages} = 500$
$Errors = 250,\,\,\,samples = 2$
Therefore,
$r = 0\,\,\,,\,\,\lambda = \dfrac{{500}}{{250}} = 2\,\,,\,\,n = 2$

Now, putting the values in formula
$P\left( {X = 0} \right) = \dfrac{{{e^{\dfrac{{ - 2}}{2}}}{\lambda ^0}}}{{0!}}$
$ \Rightarrow P\left( {X = 0} \right) = \dfrac{{{e^{ - 1}}{\lambda ^0}}}{{0!}}$
We know that if power of any positive number is zero then its value would be $1.$Also, we know that the value of $0!\,\,is\,\,1.$So, putting these values in the formula.
$ \Rightarrow P\left( {X = 0} \right) = \dfrac{{{e^{ - 1}}\left( 1 \right)}}{{\left( 1 \right)}}$
$\therefore P\left( {X = 0} \right) = {e^{ - 1}}$
Therefore, the required probability is ${e^{ - 1}}$.

Hence, the correct option is $\left( C \right)$.

Note: A Poisson distribution is a probability distribution that can be used to show how many times an event is likely to occur within a specified period of time. In other words, it is a count distribution. For example, suppose that from historical data, we know that earthquakes occur in a certain area with a rate of $2$ per month. Other than this information, the timings of earthquakes seem to be completely random. Thus, we can conclude that the Poisson process might be a very good model for earthquakes.