Question

 

A Firm studied the number of lost-time accidents occurring at its plant. Past records show that 7% of the employees suffered lost-time accidents last year. Management assumes that a special safety program will reduce these accidents to 6% in the current year. Also, it estimates that 15% of the employees who had such accidents in the last year will face a lost-time accident in the current year.

a. What is the probability an employee will experience a lost-time accident in both years (to 2 decimals)?

b. What is the probability an employee will experience a lost-time accident over the two-year period (to 2 decimals)

 

Answer 1

 

A probability is essentially a number that demonstrates the likelihood or chance that a given event will happen. Probabilities could be expressed as proportions ranging from 0 to 1. You can also express them as percentages ranging from 0% to 100%. 

Part a)

 

Let P₁ be the percent of lost time accidents in the first year.

 

P₁ = 7%

 

It’s given that 15% of the employees having an accident last year will have an accident this year (say P₂).

 

Let P₁₁ be the probability of having an accident in both years.

 

So the percent of employees having an accident in both years is:

 

P₁₁ = P₁ x P₂

 

P₁₁ = 7% x 15% = 0.07 x 0.15 = 0.0105 = 1.05%

 

1.05% is the percentage of employees having an accident in both years.

 

Part b)

 

Let P₀₀ be the probability that the employees have no accident in either year

 

P₁₀ be the probability that the employees have an accident in the first year but not the second

 

P₀₁ be the probability that the employees have an accident in the second year but not the first

 

and

 

P₁₁ be the probability that the employees have an accident in both years

 

P₁₁ (from the previous calculation) is 1.05%

 

P₁₀ + P₁₁ = 7% (total number of employees having an accident the first year),

 

so

 

P₁₀ = 7% - P₁₁ = 7% - 1.05% = 5.95%

 

We also know the total number of employees having an accident the second year is:

 

P₀₁ + P₁₁ = 6%

 

So the number of employees having an accident only in the second year P₀₁ is given by

 

P₀₁ + P₁₁ = 6%

 

P₀₁ + 1.05% = 6%

 

P₀₁ = 4.95%

 

So what is the total percentage of employees who have accidents in either year?

 

It is

 

P₁₀ + P₀₁ + P₁₁ = 5.95% + 4.95% + 1.05%

 

= 11.95%

 

(percent that had an accident in the first year but not the second + percent that had an accident in the second year but not the first + percent that had an accident in both years)

 

11.95% of all employees have had at least one accident in the 2 year period.

 

 

a) 1.05% is the percentage of employees having an accident in both years.

b) 11.95% of all employees have had at least one accident in the 2 year period.