Poisson distribution application example

Question

There's this example in a book I'm reading, and I don't understand the solution — how exactly to calculate the probability $P(X ≥ 12) $

Example: During the last few years in Gotham City, a provincial city with more than 100,000 inhabitants, there have been eight serious fires per year, on average. Last year, by contrast, twelve serious fires blazed. How exceptional is that?

Solution: To answer this question, you calculate the probability of twelve or more fires and not the probability of exactly twelve fires. The desired probability is given by $P(X ≥ 12) = 0.112$

Answer 1

It got to me.

$P(X=k)=e^{-x}\frac{x^k}{k!}$

Since all probabilities(constant $x$ and variable $k$) sum up to $1$, we can find $P(X≥12)$ by extracting from $1$ all probabilities $P(X)$ excluding those with $X≥12$

$$1 - P(X<12) = 1 - \sum_{i=0}^{11}P(i)$$

Or I could just calculate probabilities from $12$ up to $x^2$ $$P(X≥12) = \sum_{i=12}^{x^2}P(i)$$

And one thing I did not see clearly is that in the example $x=8$ because $x=np$ which is the average, so is the $8$ in the example

Answer 2

100000 is Irelavent

$X\sim$ $Po(8)$

$P(X\geq12)=1-P(X\leq 11)=1-0.8881=0.1119$