For $X \sim B(10,000, 0.001)$ (i.e., $X$ is binomially distributed with $n=10,000$ and $p = 0.001$), I need to evaluate $P(X=10)$ by using both the Laplace and Poisson theorems, respectively, and then identifying which is more accurate.
Laplace Theorem: Let $X_{i}\sim B(1,p)$ $i=1,2,\cdots$ are independent. Then, as $n \to \infty$, $$P\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\leq x \right) = \sum_{k=1}^{\lfloor{nx}\rfloor}\begin{pmatrix}n \\ k \end{pmatrix} p^{k}(1-p)^{\lfloor{nx}\rfloor-k} \longrightarrow \int_{-\infty}^{x} \frac{1}{\sqrt{2 \pi p (1-p)}}\exp{\left\{ - \frac{(x-p)^{2}}{2p(1-p)} \right\}}dx$$ for any $x$.
Poisson Theorem: Let $X_{n,i}\sim B(1, p_{n})$ for $n, i=1,2,\cdots$ be independent. Then, as $n p_{n} \to \lambda > 0$, $$ P \left(\sum_{i=1}^{n}X_{n,i}=k \right) = \begin{pmatrix}n \\k \end{pmatrix} p_{n}^{k}(1-p_{n})^{n-k} \longrightarrow \frac{\lambda^{k}e^{-\lambda}}{k!}$$ for any $k = 0,1,2,\cdots$
My major problem here is translating this into dealing with the probability that $X=10$ as opposed to the probability of a sum being less than a value. Also, I have seen an example where we were trying to find $P(X \leq 70)$, but I don't know how to do it for equality. Could somebody please help me figure these out? Thank you.