Question

Problems at major companies. The Organization Development Journal (Summer 2006) reported on a survey of human resource officers (HROs) at major employers. The focus of the study was employee behaviour, namely, absenteeism and turnover. The study found that 55% of the HROs had problems with employee absenteeism; 41% had problems with turnover. Suppose that 22% of the HROs had problems with both absenteeism and turnover. Use this information to find the probability that an HRO selected from the group surveyed had problems with either employee absenteeism or employee turnover.

Short answer

0.78

Step-by-step solution

Introduction

The probability of an event is a measure of the likelihood of an event occurring when an experiment is performed. When there are just two possible outcomes, complementary occurrences occur. There is another event${\mathbf{A}}^{\mathbf{c}}$ for every event A, which ${\mathbf{A}}^{\mathbf{c}}$signifies the complimentary event.

$\mathbf{P}\mathbf{\left(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{–}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{.}$

Determine the probability that an HRO was chosen from the sample who experienced issues with either employee absenteeism or employee turnover

Let’s take A1 as the HRO problem with employee absenteeism and A2 as the HRO problem with employee turnover.

Therefore, we get:

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{55}\mathbf{\%} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{55}}{\mathbf{100}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{55} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{41}\mathbf{\%} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{41}}{\mathbf{100}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{41} \end{gathered}$$

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{1}\mathbf{\cap }\mathbf{A}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{22}\mathbf{\%}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{22}}{\mathbf{100}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{22}\end{array}$

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{55}\mathbf{\%}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{55}}{\mathbf{100}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{55}\end{array}$

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{41}\mathbf{\%}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{41}}{\mathbf{100}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{41}\end{array}$

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{1}\mathbf{\cap }\mathbf{A}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{22}\mathbf{\%}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{22}}{\mathbf{100}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{22}\end{array}$

Hence, the probability that HRO had a problem with either employee absenteeism or employee turnover is:

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{either}\mathbf{}\mathbf{employee}\mathbf{}\mathbf{absenteeism}\mathbf{}\mathbf{or}\mathbf{}\mathbf{}\mathbf{employee}\mathbf{}\mathbf{turn}\mathbf{}\mathbf{over}\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{–}\mathbf{P}\mathbf{(}\mathbf{neither}\mathbf{}\mathbf{employee}\mathbf{}\mathbf{absenteeism}\mathbf{}\mathbf{or}\mathbf{}\mathbf{employee}\mathbf{}\mathbf{turn}\mathbf{}\mathbf{over}\mathbf{)}\\ \mathbf{=}\mathbf{1}\mathbf{–}\mathbf{0}\mathbf{.}\mathbf{22}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{78}\end{array}$

Therefore, the required probability is 0.78.