The safety limit of a crane is 42 tons, the mean weight and the standard deviation of a large number of iron rods are 0.2 ton and 0.5 ton respectively 20 rods are lifted at a time. Find the probability of an accident.

To determine the probability of an accident when lifting 20 iron rods with a crane that has a safety limit of 42 tons, we can use the concepts of probability and statistics, particularly the normal distribution. Let's break this down step by step.

Understanding the Problem

We know the following:

• The safety limit of the crane is 42 tons.
• The mean weight of an iron rod is 0.2 tons.
• The standard deviation of the weight of the rods is 0.5 tons.
• We are lifting 20 rods at a time.

Calculating the Total Weight of the Rods

When lifting 20 rods, the total weight can be modeled as the sum of the weights of the individual rods. Since the weights of the rods are normally distributed, the total weight will also be normally distributed. We can calculate the mean and standard deviation of the total weight as follows:

• Mean of total weight: The mean weight of 20 rods is:

Mean = Number of rods × Mean weight of one rod = 20 × 0.2 = 4 tons.

• Standard deviation of total weight: The standard deviation of the sum of independent random variables is the square root of the sum of their variances. Since the variance is the square of the standard deviation:

Standard deviation = √(Number of rods) × Standard deviation of one rod = √20 × 0.5 ≈ 2.236 tons.

Finding the Probability of Exceeding the Safety Limit

Next, we need to find the probability that the total weight of the 20 rods exceeds the crane's safety limit of 42 tons. We can standardize this using the Z-score formula:

Z = (X - μ) / σ

Where:

• X: The value we are comparing to (42 tons).
• μ: The mean of the total weight (4 tons).
• σ: The standard deviation of the total weight (approximately 2.236 tons).

Substituting the values:

Z = (42 - 4) / 2.236 ≈ 17.0

Interpreting the Z-score

A Z-score of 17.0 is extremely high, indicating that the total weight of 20 rods is far below the safety limit of the crane. In fact, in a standard normal distribution, a Z-score this high corresponds to a probability that is practically zero. This means that the likelihood of the total weight of the rods exceeding 42 tons is virtually nonexistent.

Final Thoughts

In conclusion, the probability of an accident occurring due to exceeding the crane's safety limit when lifting 20 iron rods is extremely low, essentially zero. This analysis shows how statistical methods can be applied to assess safety in practical scenarios, ensuring that operations remain within safe limits.