Question

A manufacturing concern employing a large number of workers finds that, over a period of time, the average absentee rate is 2 workers per shift. The probability that exactly 2 workers will be absent in a chosen shift at random is

Answer 1

Hint: According to the question we have to determine the probability that exactly 2 workers will be absent in a chosen shift at random when a manufacturing concern employing a large number of workers finds that, over a period of time, the average absentee rate is 2 workers per shift. So, first of all we have to use the Poisson distribution rule as explained below:
Poisson distribution rule: The Poisson distribution is the discrete probability distribution of the number of given events occurring in a given time period, given the average number of times the event occurs over that time period.

Formula used: $\Rightarrow P(x;\mu )={\displaystyle \frac{e^{-1}{\mu }^x}{x!}}...............(A)$
Where, x is the actual number of successes that occur in a specified region, $\mu$is the mean number of successes that occur in a specified region, and $P(x;\mu )$is the Poisson probability that exactly x success occurs in a Poisson experiment.

Complete step-by-step solution:
Step 1: First of all we have to determine the mean number of successes is $\mu$and as in the question $\mu$= 2 and x = 2 (number of 2 workers)
Step 2: Now, we have to substitute all the values in the formula (A) as mentioned in the solution hint. Hence,
$\Rightarrow P(2;2)={\displaystyle \frac{e^{-2}{2}^2}{2!}}$
Step 3: Now, to obtain the values of expression we have to solve the expression (1) as obtained in the solution step 2. Hence,
$$\begin{gathered} \Rightarrow P(2;2)={\displaystyle \frac{e^{-2}2\times 2}{2\times 1}} \\ \Rightarrow P(2;2)=2e^{-2} \end{gathered}$$

Final solution: Hence, with the help of the formula (A) as mentioned in the solution hint we have obtained the probability that exactly 2 workers will be absent in a chosen shift at random is $P(2;2)=2e^{-2}$

Note: The Poisson distribution is a discrete probability for the counts of events that occur randomly in a given interval of time and we have to let X as the number of events in a given interval and then if the mean number of events per interval is$\mu$.
In Poisson distribution e is a mathematical constant where, $e=2.718282$ and $\mu$ is the parameter of the distribution.