Chapter 5, Problem R5.7RE

(a)

To determine

To draw:

Tree diagram for the given data

Explanation of Solution

Given information:

Percentage of employees who use drugs $=\text{ 4}\%$

Percentage of employees whose drug test gave false positive result $=\text{ 5}\%$

Percentage of employees whose drug test gave false negative result $=\text{ 1}0\%$

Let the event of “employees who take illegal drugs” be denoted by D.

The event of “employees who do not take illegal drugs” be denoted by ND.

The event of “employees whose result was positive” be denoted by P.

The event of “employees whose result was negative” be denoted by N.

Graph:

The tree diagram is

  

(b)

To determine

To calculate: Probability that the drug test result positive

Answer to Problem R5.7RE

Probability that the drug test result positive is 0.084.

Explanation of Solution

Formula used:

  $\text{P (P) = P }\left(\text{D }\cap \text{ P}\right)+\text{P }\left(\text{ND }\cap \text{ P}\right)$

Calculation:

From the tree diagram in the sub part a,

  $\text{P }\left(\text{D }\cap \text{ P}\right)\text{ = 0}\text{.04 }\times \text{ 0}\text{.90 = 0}\text{.036}$

  $\text{P }\left(\text{ND }\cap \text{ P}\right)\text{ = 0}\text{.96 }\times \text{ 0}\text{.05 = 0}\text{.048}$

  $\text{P }\left(\text{P}\right)=0.0\text{36 }+0.0\text{48 }=0.0\text{84}$

(c)

To determine

To calculate: Probability that an employee uses illegal drugs given the result is positive.

Answer to Problem R5.7RE

Probability that an employee uses illegal drugs given the result is positive is 0.42857.

Explanation of Solution

Formula used:

  $\text{P (D | P) = }\frac{\text{P }\left(\text{D }\cap \text{ P}\right)}{\text{P (P)}}$

Calculation:

From the sub part b, $\text{P }\left(\text{P}\right)=0.0\text{84}$

  $\text{P (D }\cap \text{ P)}=0.0\text{36}$

  $\text{P (D | P ) = }\frac{0.036}{0.084}=0.42857$