Question

A company studied the number of lost-time accidents occurring at its Brownsville, Texas, plant. Historical records show that 6% of the employees suffered lost-time accidents last year. Management believes that a special safety program will reduce such accidents to 5% during the current year. In addition, it estimates that 15% of employees who had lost-time accidents last year will experience a lost-time accident during the current year.

Round your answers to three decimal places.

a. What percentage of the employees will experience lost-time accidents in both years?


b. What percentage of the employees will suffer at least one lost-time accident over the two-year period?

Answer 1

[Section: Concepts and reason}

Event: The collection or the set of outcomes in an experiment is called as an event.

Union of two events: The set of the outcomes that belong to either the two events or any of the two events is called as union of two events.

Intersection of two events: The set of the outcomes that belong to both the two events s is called as intersection of two events.

Probability: The ratio of the number of favorable outcomes to certain event and total number of possible outcomes is called as the probability of an event.

Fundamentals

The probability of an event is defined as,

${\rm{Probability}} = \frac{{{\rm{Number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}\,{\rm{for}}\,{\rm{an}}\,{\rm{event}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes}}}}$

Define A and B as two events,

The addition rule for the probability is, $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$

One of the rules for the probability is,

• $P\left( {A \cap B} \right) = P\left( {A|B} \right) \times P\left( B \right)$

(a)

The percentage of the employees will experience lost-time accidents in both years are obtained below:

Let L is denoted as the event that employees suffer lost-time accidents during the last year and C is denoted as the event that employees suffer lost-time accidents during the current year

From the information given, the 6% of the employees suffered lost-time accidents last year. That is, $P\left( L \right) = 0.06$ , 6% of the employees suffered lost-time accidents current year. That is, $P\left( C \right) = 0.05$ , and 15% of employees who had lost-time accidents last year will experience a lost-time accident during the current year. That is, $P\left( {C|L} \right) = 0.15$ .

The required probability is,

$\begin{array}{c}\\P\left( {L \cap C} \right) = {\rm{P}}\left( {C|L} \right) \times P\left( L \right)\\\\ = 0.15 \times 0.06\\\\ = 0.009\\\\ = 0.9\% \\\end{array}$

(b)

The percentage of the employees will suffer at least one lost-time accident over the two-year period is obtained below:

The required probability is,

$\begin{array}{c}\\P\left( {L \cup C} \right) = P\left( L \right) + P\left( C \right) - P\left( {L \cap C} \right)\\\\ = 0.06 + 0.05 - 0.009\\\\ = 0.101\\\\ = 10.1\% \\\end{array}$

Ans: Part a

The percentage of the employees will experience lost-time accidents in both years are 0.9%.

Part b

The percentage of the employees will suffer at least one lost-time accident over the two-year period is 10.1%.