Question
Suppose that asteroid impacts on the earth are modelled with a Poisson distribution. Suppose large asteroids (of diameter 1km or more) are estimated to impact the Earth once every 500,000 years. How many years are required for the probability of at least one 1km diameter impact to exceed 50%?
Using the information provided above I used the following formula:
$P(n, t)=\frac{e^{-λt}(λt)^n}{n!}$
For the values: $λ=\frac{1}{500000}$ and $n=1$
Giving the equation: $0.5=\frac{e^{-\frac{1}{500000}t}(\frac{1}{500000}t)^1}{1!}$
But after trying to solve for t, I could not isolate it. I am unsure now if this is the right approach. What am I not doing correctly?
Here's a general tip: if the number of times an event occurs can be modeled with a Poisson distribution with parameter $\lambda$, the probability distribution of the waiting time $T$ for the next time the event occurs can be modeled as $T \sim$ an exponential distribution with parameter $\lambda$. From there, it's a matter of figuring out the upper bound $b$ of the integral of the PDF so that the integral from $0 \to b = 0.5$