Question

A certain factory operates three different shifts. Over the last year, 200 accidents have occurred at the factory. Some of these can be attributed at least in part to unsafe working conditions, whereas the others are unrelated to working conditions. The accompanying table gives the percentage of accidents falling in each type of accident–shift category.

Unsafe Unrelated

Conditions to Conditions

Day 10% 35%

Shift Swing 8% 20%

Night 5% 22%

Suppose one of the 200 accident reports is randomly selected from a file of reports, and the shift and type of accident are determined.

a. What are the simple events?

b. What is the probability that the selected accident was attributed to unsafe conditions?

c. What is the probability that the selected accident did not occur on the day shift?

Short answer

a.\(S = \left\{ {\left\{ {A,D} \right\},\left\{ {A,S} \right\},\left\{ {A,N} \right\},\left\{ {B,D} \right\},\left\{ {B,S} \right\},\left\{ {B,N} \right\}} \right\}\)

b. The Probability that the selected student was attributed to unsafe conditions is 0.23

c. The probability that the selected accident did not occur on the day shift is 0.55

Step-by-step solution

Given information

The percentages of the accidents falling in a different type of accident-shift category are provided in a table.

The total number of accidents are 200.

Describe the simple events

a.

Let D be the event representing a day shift.

Let S be the event representing the Swing shift.

Let N be the event representing the Night shift.

Let A be the event representing the unsafe conditions.

Let B be the event representing the unrelated to conditions.

The above events can occur together as well.

The simple events are provided as,

\(S = \left\{ {\left\{ {A,D} \right\},\left\{ {A,S} \right\},\left\{ {A,N} \right\},\left\{ {B,D} \right\},\left\{ {B,S} \right\},\left\{ {B,N} \right\}} \right\}\)

Compute the probability

b.

Using the provided table,

The probability that the selected accident was attributed to unsafe conditions is computed as,

\(\begin{aligned}P\left( A \right) &= P\left( {A \cap D} \right) + P\left( {A \cap S} \right) + P\left( {A \cap N} \right)\\ &= 0.10 + 0.08 + 0.05\\ &= 0.23\end{aligned}\)

Therefore, the probability that the selected accident was attributed to unsafe conditions is 0.23.

c.

The probability that the selected accident occur on the day shift is computed as,

\(\begin{aligned}P\left( D \right) &= P\left( {A \cap D} \right) + P\left( {B \cap D} \right)\\ &= 0.10 + 0.35\\ &= 0.45\end{aligned}\)

The probability that the selected accident did not occur on the day shift is computed as,

\(\begin{aligned}P\left( {D'} \right) &= 1 - P\left( D \right)\\ &= 1 - 0.45\\ &= 0.55\end{aligned}\)

Therefore, the probability that the selected accident did not occur on the day shift is 0.55.