## Introduction

As a website operator, it’s essential to have a comprehensive understanding of various subjects to provide adequate assistance and support to users. One of the most critical fields is probability, which involves understanding the likelihood of certain events occurring. In this article, we will be looking at one of the probability homework exercises and providing the answer key.

## The Problem

The third homework exercise in unit 12 focuses on geometric probability. The question states that a light bulb manufacturer produces light bulbs with a lifespan of 300 hours. The duration that the bulb will last is distributed exponentially. It states that a distributor has a box of 100 bulbs, and all boxes have the same average lifespan of 300 hours.

The question then asks for the probability that the 50th bulb from the box will last less than 200 hours.

## The Calculation

Geometric probability deals with probabilities that are created when two random variables interact. In this case, we have the position of the bulb in the box (50th) and its corresponding lifespan. The formula for geometric probability is P(A) = 1 – ((1-p)n), where p is the probability of the event happening, and n is the number of trials.

To calculate the probability of the 50th light bulb lasting less than 200 hours, we need to determine the success rate of the bulbs with a lifespan of 200 hours or less. We know that the light bulbs distributed by the manufacturer have an exponential distribution, where the mean lifespan is 300 hours.

The probability of a bulb lasting less than 200 hours can be calculated by finding the area under the curve of the exponential function between 0 and 200 hours. Using the formula for the cumulative distribution function of an exponential distribution, we can calculate the probability of a bulb lasting less than 200 hours to be approximately 0.393.

Therefore, using the formula for geometric probability, we can calculate the probability of the 50th bulb lasting less than 200 hours to be P(A) = 1 – ((1-0.393)50) = 0.999999999998509. This answer can be rounded up to 1, which means there is a 100% chance that the 50th bulb from the box will last less than 200 hours.

## Conclusion

In conclusion, probability is an essential field that website operators need to understand. In this article, we have looked at the third homework exercise in unit 12 of geometric probability and provided the answer key. From our calculations, we can determine that there is a 100% chance that the 50th light bulb in the box will last less than 200 hours. As a website operator, having a clear understanding of probability will enable you to provide valuable assistance and support to your users.