I am told that the waiting time between now and the moment when the next volcano erupts follows an exponential distribution $$f(t) = \lambda e^{-\lambda t}$$ with parameter $\lambda = 0.1$ events per year. Let $X$ be the random variable representing the waiting time between now and the next eruption. My understanding is that in order to calculate the probability of the next volcano happening within the next 50 years is given by $$P(X<50) = \int_{0}^{50}e^{-\lambda x}dx = -(e^{-50\lambda}-1)\approx 0.99. $$ Now, I am asked to compute the probability that at most 2 volcanoes will erupt within the next 50 years. How do I accomplish this?
What I have tried is doubling the rate $\lambda$ to $2\lambda = 0.2$ for that would be the rate of 2 events per unit time. I obtained $P(X<50) = 0.99995$ which to me seems unrealistic. I don't know if this is a justifiable move. I don't have the answer to the problem, I just thought about it as I am studying for a midterm exam.
[self-study]
tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$