# Probability and lifetime of a lightbulb

• SUchica10
In summary, Jacinda10, the probability that a lightbulb fails within the first 200 hours is 0.2%. The probability that it fails after 200 hours but before 800 hours is 0.8%. The median lifetime of these lightbulbs is approximately 800 hours.
SUchica10
This probability problem is for my calculus class...

a) A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean = 1000. Use this model to find the probability that a bulb
(i) fails within the first 200 hours
(ii) burns for more than 800 hours

b) What is the median lifetime of these lightbulbs?

Show us some work. You must have learned something about probability density functions in your class that is just enough to get started.

My teacher does not explain well and I don't know where to start... I am thinking that I will need to integrate but I am not sure what the function would be.

It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean = 1000.

What does this tell you off the top of your head? Try graphing what such a function must look like

Why don't you try standard deviation? Or the standard-normal curve? That is how I would go about solving this...

u know that the area under the curve must be 1
and the mean must be 1000

now make a friken model.

jacinda10 i don't think this is normally distributed

jacinda10 said:
Why don't you try standard deviation? Or the standard-normal curve? That is how I would go about solving this...
In spite of the fact that the problem says it is an exponential distribution?

SUchica10, as I just said, you are told that the distribution is exponential. Have you looked up "exponential distribution" in your textbook?

The way to handle any question like this , no matter the distribution (well there are some pathological examples but they are rare) is to integrate. So integrate under that probability distribution from 0 to 200 and then 800 to infinity. As long as you have normalized the distribution the result is the probability you seek.

## 1. What is the relationship between probability and the lifetime of a lightbulb?

The probability of a lightbulb burning out is directly related to its lifetime. A higher probability means the lightbulb is more likely to burn out sooner, while a lower probability means it is more likely to last longer. This is because probability measures the likelihood of an event occurring, and in this case, the event is the lightbulb burning out.

## 2. How is probability calculated for the lifetime of a lightbulb?

Probability is calculated by dividing the number of times a certain outcome (such as a lightbulb burning out) occurs by the total number of possible outcomes. In the case of a lightbulb, this would involve collecting data on the lifetimes of multiple lightbulbs and calculating the percentage of those that burned out within a certain time frame.

## 3. What factors can affect the probability of a lightbulb burning out?

There are several factors that can affect the probability of a lightbulb burning out, including the quality of the lightbulb, the conditions in which it is used (e.g. temperature, humidity), and the frequency of use. A higher quality lightbulb, for example, may have a lower probability of burning out compared to a lower quality one.

## 4. How can probability be used to predict the lifetime of a lightbulb?

Probability can be used to make predictions about the lifetime of a lightbulb, but it is not a guarantee. By calculating the probability of a lightbulb burning out within a certain time frame, one can estimate the likelihood of it lasting for a specific duration. However, other unforeseen factors may also impact the actual lifetime of the lightbulb.

## 5. Can probability be used to extend the lifetime of a lightbulb?

Probability cannot directly extend the lifetime of a lightbulb, as it is simply a measure of likelihood. However, by understanding the factors that can affect the probability of a lightbulb burning out, one can take steps to increase the chances of the lightbulb lasting longer. For example, using the lightbulb in optimal conditions and handling it carefully can potentially extend its lifetime.

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