Question

Work-related accidents at a construction site tend to have a Poisson distribution with an average of 2 accidents per week. a. What is the probability that there will be no work-related accidents at this site during a given week? b. What is the probability that there will be at least 1 work-related accident during a given week? c. What is the distribution of the number of work-related accidents at this site per month? d. What is the probability that there will be no work-related accidents during a given month?

Short answer

In summary, work-related accidents at the construction site follow a Poisson distribution. The probability of not having any accidents in a week is approximately 13.53%, while the probability of having at least one accident during a week is approximately 86.47%. The distribution of accidents per month also follows a Poisson distribution with an average of 8 accidents a month. The probability of having no work-related accidents in a month is very low, at approximately 0.0335%.

Step-by-step solution

Part a: Probability of no accidents in a week

To find the probability of no work-related accidents in a given week, we will use the Poisson distribution formula where λ = 2 and x = 0. P(0) = (e^(-2) * 2^0) / 0! Calculating the probability: P(0) = (0.135335 * 1) / 1 = 0.135335 So, the probability of having no work-related accidents during a given week is approximately 13.53%.

Part b: Probability of at least 1 accident in a week

To find the probability of at least one work-related accident in a given week, we will calculate the complementary probability of having no accidents. P(at least 1) = 1 - P(0) Using the result from Part a: P(at least 1) = 1 - 0.135335 ≈ 0.864665 So, the probability of having at least one work-related accident during a given week is approximately 86.47%.

Part c: Distribution of accidents per month

In this part, we need to find the distribution of accidents per month. To do this, we need to find the new average rate of occurrence (λ) for a month. Since there is an average of 2 accidents per week, and there are 4 weeks in a month (approximately), we will use: λ per month = 2 accidents/week * 4 weeks/month = 8 accidents/month So, the distribution of work-related accidents per month also follows a Poisson distribution with an average of 8 accidents per month.

Part d: Probability of no accidents in a month

To find the probability of no work-related accidents in a given month, we will use the Poisson distribution formula with the new average rate of occurrence, λ = 8 (accidents per month), and x = 0. P(0) = (e^(-8) * 8^0) / 0! Calculating the probability: P(0) = (0.000335 * 1) / 1 ≈ 0.000335 So, the probability of having no work-related accidents during a given month is approximately 0.0335%.

Key concepts

Work-Related Accidents Statistics

Understanding work-related accidents statistics involves discerning how often accidents occur within a workplace setting over a specified period. The rate of these accidents can often be modeled by probability distributions. In our scenario, accidents at a construction site follow a Poisson distribution, which is particularly useful for estimating the probability of a given number of events occurring within a fixed interval of time or space. The Poisson distribution is characterized by its rate parameter, denoted as \( \lambda \) (lambda), which in this case is 2 accidents per week.

By analyzing such statistics, workplace safety programs can be designed and implemented to reduce the frequency and severity of these incidents, aiming to create a safer environment for employees.

Probability Calculations

Probability calculations are essential for understanding various risk factors and for making informed decisions. When calculating the probability of an event in a Poisson distribution, the formula \( P(x) = (e^{-\lambda} \cdot \lambda^x) / x! \) is used. In this formula, \( e \) represents the base of the natural logarithm, \( \lambda \) is the rate of occurrence, \( x \) is the number of events (accidents), and \( x! \) is the factorial of \( x \).

With the provided statistics, such as an average of 2 accidents per week, we can calculate the probability of different scenarios, like no accidents occurring in a week, or at least one accident occurring. This understanding of probability calculations empowers students and professionals alike to interpret statistical data effectively.

Distribution of Accidents

The distribution of accidents in a given context offers insights into patterns and frequencies of these occurrences. When working with distributions like the Poisson distribution, it's crucial to consider the time frame in question. As with the transition from a weekly to a monthly perspective, we recalculated the average rate (\( \lambda \) parameter) to reflect 8 accidents per month, assuming an average of 4 weeks per month.

The nature of the Poisson distribution helps us understand that even if accidents are random, they are not without predictable patterns when viewed over a longer period. This helps in resource allocation, such as the number of safety inspectors needed over time.

Complementary Probability

Complementary probability is an essential concept in probability theory, used to determine the likelihood of an event not happening by subtracting the probability of the event from 1. It is especially helpful in scenarios where calculating the probability of non-occurrence directly is complex.

For example, if we know that the probability of having no work-related accidents in a week is approximately 13.53%, we can say the probability of having at least one accident is the complement of this, calculated as 1 - 0.135335, which gives us approximately 86.47%. Understanding the use of complementary probabilities is crucial for risk assessment and management in real-world situations.