Study of why EMS workers leave the job. An investigation into why emergency medical service (EMS) workers leave the profession was published in the Journal of Allied Health (Fall 2011). The researchers surveyed a sample of 244 former EMS workers, of which 127 were fully compensated while on the job, 45 were partially compensated, and 72 had no compensated volunteer positions. EMS workers who left because of retirement were 7 for fully compensated workers, 11 for partially compensated workers, and 10 for no compensated volunteers. One of the 244 former EMS workers is selected at random.

a. Find the probability that the former EMS worker was fully compensated while on the job.

b. Find the probability that the former EMS worker was fully compensated while on the job and left due to retirement.

c. Find the probability that the former EMS worker was not fully compensated while on the job.

d. Find the probability that the former EMS worker was either fully compensated while on the job or left due to retirement.

Step by step solution

01

Introduction

Probability quantifies the uncertainty or confidence of an event's occurrence.

$\mathbf{Probability}\mathbf{=}\frac{\mathbf{Favourable}\mathbf{}\mathbf{out}\mathbf{}\mathbf{come}}{\mathbf{Total}\mathbf{}\mathbf{out}\mathbf{}\mathbf{come}}$

02

Determine the probability that the former EMS worker was fully compensated

$\begin{array}{l}\mathbf{Totaloutcome}\mathbf{=}\mathbf{244}\\ \mathbf{A}\mathbf{\left(}\mathbf{fullycompensated}\mathbf{\right)}\mathbf{=}\mathbf{127}\\ \mathbf{B}\mathbf{\left(}\mathbf{Partiallycompensated}\mathbf{\right)}\mathbf{=}\mathbf{45}\\ \mathbf{C}\mathbf{(}\mathbf{non}\mathbf{-}\mathbf{compensated}\mathbf{)}\mathbf{=}\mathbf{72}\end{array}$

$\mathbf{D}\mathbf{(}\mathbf{left}\mathbf{}\mathbf{due}\mathbf{}\mathbf{to}\mathbf{}\mathbf{retirement}\mathbf{)}\mathbf{=}\mathbf{7}\mathbf{+}\mathbf{11}\mathbf{+}\mathbf{10}\mathbf{=}\mathbf{28}$

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{fully}\mathbf{}\mathbf{compensated}\mathbf{)}\mathbf{=}\frac{\mathbf{Favourable}\mathbf{}\mathbf{out}\mathbf{}\mathbf{come}}{\mathbf{Total}\mathbf{}\mathbf{out}\mathbf{}\mathbf{come}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{127}}{\mathbf{244}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{53} \end{gathered}$$

Hence, the required probability is 0.53.

03

Determine the probability that the former EMS worker was fully compensated and left due to retirement

$\mathbf{P}\mathbf{(}\mathbf{fully}\mathbf{}\mathbf{compensated}\mathbf{}\mathbf{and}\mathbf{}\mathbf{left}\mathbf{}\mathbf{due}\mathbf{}\mathbf{to}\mathbf{}\mathbf{retirement}\mathbf{)}\mathbf{=}\frac{\mathbf{7}}{\mathbf{244}}$

Hence, the required probability is 7/244.

04

Determine the probability that the former EMS worker was not fully compensated

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{notfullycompensated}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{P}\mathbf{\left(}\mathbf{fullycompensated}\mathbf{\right)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{127}}{\mathbf{244}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{244}\mathbf{-}\mathbf{127}}{\mathbf{244}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{117}}{\mathbf{244}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{48}\end{array}$

Hence, the required probability is 0.48.

05

Determine if the former EMS worker was fully compensated or left due to retirement

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{fully}\mathbf{}\mathbf{compensated}\mathbf{}\mathbf{or}\mathbf{}\mathbf{left}\mathbf{}\mathbf{due}\mathbf{}\mathbf{to}\mathbf{}\mathbf{retirement}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{fully}\mathbf{}\mathbf{compensated}\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}\mathbf{left}\mathbf{}\mathbf{due}\mathbf{}\mathbf{to}\mathbf{}\mathbf{retirement}\mathbf{)}\\ \mathbf{-}\mathbf{P}\mathbf{(}\mathbf{fully}\mathbf{}\mathbf{compensated}\mathbf{}\mathbf{and}\mathbf{}\mathbf{left}\mathbf{}\mathbf{due}\mathbf{}\mathbf{to}\mathbf{}\mathbf{retirement}\mathbf{)}\\ \mathbf{=}\frac{\mathbf{127}}{\mathbf{244}}\mathbf{+}\frac{\mathbf{28}}{\mathbf{244}}\mathbf{-}\frac{\mathbf{7}}{\mathbf{244}}\\ \mathbf{=}\frac{\mathbf{127}\mathbf{+}\mathbf{28}\mathbf{-}\mathbf{7}}{\mathbf{244}}\\ \mathbf{=}\frac{\mathbf{148}}{\mathbf{244}}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{61}\end{array}$

Hence, the required probability is 0.61.

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