Calculating Probabilities of Light-Bulb Lifetimes Using Normal Distribution

Understanding the expected lifetime of a light-bulb can be crucial for various applications, such as predicting maintenance needs and optimizing product design. In this context, we will explore the probability that a randomly selected light-bulb has a lifetime between 3000 and 3100 hours, assuming the lifetimes follow a normal distribution with a mean of 3000 hours and a standard deviation of 200 hours.

Problem Statement

The lifetime of a light-bulb, denoted by X, is normally distributed with a mean (μ) of 3000 hours and a standard deviation (σ) of 200 hours. The task is to find the probability that a randomly selected light-bulb has a lifetime between 3000 and 3100 hours. This can be mathematically written as:

P(3000 X 3100) ?

Step-by-Step Solution

To solve this problem, we will use the standardization process, which converts the variable X into a standard normal variable (Z).

Standardization

The standard normal variable Z is defined as:

Z

Using this transformation, we can rewrite the probability statement as follows:

P(3000 X 3100) P(

Using the Standard Normal Distribution Table

The standard normal distribution table provides the probabilities associated with various values of Z. From the table, we can find the cumulative probabilities for Z values of 0 and 0.5. The cumulative probability for Z 0.5 is approximately 0.6915, and the cumulative probability for Z 0 is 0.5.

Thus, the probability that Z lies between 0 and 0.5 can be calculated as:

P(0 Z 0.5) 0.6915 - 0.5 0.1915

Conclusion

Therefore, the probability that a randomly selected light-bulb has a lifetime between 3000 and 3100 hours is 0.1915 or 19.15%.

Further Reading

For more detailed information about normal distribution and how it is used in various applications, you can refer to the following resources:

Understanding how to manipulate and interpret standard normal variables is crucial for solving real-world problems involving normal distributions. By standardizing random variables, we can leverage the standard normal distribution to determine probabilities, making the analysis process more straightforward and efficient.