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The distance $({\text{in }}km)$ of $40$ engineers from their residence to their workplace were found as follows:
$\begin{array}{*{20}{c}}
  5&3&{10}&{20}&{25}&{11}&{13}&7&{12}&{31} \\
  {19}&{10}&{12}&{17}&{18}&{11}&{32}&{17}&{16}&2 \\
  7&9&7&8&3&5&{12}&{15}&{18}&3 \\
  {12}&{14}&2&9&6&{15}&{15}&7&6&{12}
\end{array}$

What is the empirical probability that an engineer lives:
i) Less than $7{\text{ }}km$ from her place of work?
ii) More than or equal to $7{\text{ }}km$ from her place of work?
iii) Within $\dfrac{1}{2}{\text{ }}km$ from her place of work?

Answer Verified Verified
Hint: Here empirical probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trails, not in a theoretical sample space but in an actual experiment.

Complete step-by-step answer:
Given that total number of engineers $ = 40$
From the above data it is clear that,
Number of engineers who live at a distance of less than $7{\text{ }}km$ from their place of work $ = 9$
Number of engineers who live at a distance which is more than or equal to $7{\text{ }}km$ from their place of work $ = 40 - 9 = 31$
Number of engineers living within $\dfrac{1}{2}{\text{ }}km$ from their place of work $ = 0$
$
P({\text{engineer lives less than }}7km{\text{ from her place of work) =
}}\dfrac{{{\text{number of engineers living less than }}7km{\text{ from their place of work}}}}{{{\text{total number of engineers}}}} \\
  {\text{ = }}\dfrac{9}{{40}} \\
 $

$
P({\text{engineer lives more than or equal }}7km{\text{ from her place of work) =
}}\dfrac{{{\text{number of engineers living more than or equal to }}7km{\text{ from their place of work}}}}{{{\text{total number of engineers}}}} \\
  {\text{ = }}\dfrac{{31}}{{40}} \\
 $

$
P({\text{engineer lives less than }}\dfrac{1}{2}km{\text{ from her place of work) =
}}\dfrac{{{\text{number of engineers living less than to }}\dfrac{1}{2}km{\text{ from their place of work}}}}{{{\text{total number of engineers}}}} \\
  {\text{ = }}\dfrac{0}{{40}} \\
    \\
  {\text{ = 0}} \\
 $

Hence the empirical probability that an engineer lives
i) Less than $7{\text{ }}km$ from her place of work $ = \dfrac{9}{{40}}$
ii) More than or equal to $7{\text{ }}km$ from her place of work $ = \dfrac{{31}}{{40}}$
iii) Within $\dfrac{1}{2}{\text{ }}km$ from her place of work $ = 0$

Note: The probability of an event $E$ always obeys the condition $0 \leqslant P(E) \leqslant 1$. And also, the total number of outcomes in an event is always less than the total number of outcomes is the sample space.
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