Probability about lifetime of 100 bulbs (exponential distribution)

Question

I have some doubts about the following problem:

I have 100 bulbs with a lifetime represented by an exponential distribution, with an expected value of 1000 hours. Find the probability that, at least one bulb, blown down after at most 500 hours.

I have calculated the probability about one bulb with this method:

$P(X \leq 500)=\int_{0}^{500}\lambda e^{-\lambda x}dx = 1-e^{\frac{1}{2}} = 0.394$

now, how can I extend this method for all the 100 bulbs? A step-by-step solution is really appreciated, I'm really newbie about statistics/probability arguments.

Thank you so much and best regards.

EDIT: $\frac{1}{\lambda}=1000$ hours so $ \lambda = \frac{1}{1000} $

Answer 1

I assume that it means "after at most 500 hours" right? In that case your computation makes sense for one bulb. What is $\lambda$ btw?

For the second part, we may assume that the bulbs are all independent and blow down within $500$ hours with a probability of $p=0.394$. You have $100$ bulbs. What is the chance that none of these blows down?