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.
1 What percentage of the employees will experience lost-time accidents in both years?
2 What percentage of the employees will suffer at least one lost-time accident over the two-year period?
The correct answer and explanation is :
To solve this problem, we’ll define some probabilities and use them to compute the required percentages.
Given:
- P(A) = Probability of a lost-time accident last year = 6% = 0.06
- P(B) = Probability of a lost-time accident this year = 5% = 0.05
- P(B | A) = Probability of a lost-time accident this year given one occurred last year = 15% = 0.15
1. Percentage of employees with accidents in both years
This is the joint probability:
$$
P(A \cap B) = P(B | A) \cdot P(A) = 0.15 \cdot 0.06 = 0.009 = 0.9\%
$$
✅ Answer: 0.9% of employees had lost-time accidents in both years.
2. Percentage of employees with at least one accident in the two-year period
We want the probability that an employee had either a lost-time accident last year, this year, or both:
$$
P(A \cup B) = P(A) + P(B) – P(A \cap B)
$$
$$
= 0.06 + 0.05 – 0.009 = 0.101 = 10.1\%
$$
✅ Answer: 10.1% of employees had at least one accident over the two years.
Explanation (300 words)
This problem uses basic probability principles to analyze workplace safety data over two years. We are given the percentage of employees who experienced lost-time accidents in the previous year (6%), the expected percentage for the current year (5%), and the conditional probability that someone who had an accident last year will also have one this year (15%).
To determine how many employees had accidents in both years, we use the multiplication rule for conditional probability. The rule states that the probability of two events A and B both happening is equal to the probability of A times the probability of B given A. Here, event A is having an accident last year, and B is having one this year. The result, 0.9%, represents employees who were injured in both years.
Next, we calculate the percentage of employees who had at least one accident in the two-year period using the addition rule for probabilities. This rule helps us avoid double-counting individuals who had accidents in both years. By adding the probabilities of accidents in each year and subtracting the overlap, we get a more accurate total.
This analysis is important in risk management. While 5–6% annual accident rates might seem small, the cumulative risk over time becomes significant, affecting workforce productivity and safety costs. Knowing how many employees suffer repeated injuries (0.9%) can help the company tailor its safety program to high-risk individuals and further reduce incidents. Understanding these probabilities helps companies make informed decisions about the effectiveness of safety interventions.