Suppose $X$ is a random variable which follows a Poisson distribution, such that, for some positive integer $m$,
$$X \sim Po(0.01m)$$
Find the least value of $m$ such that
$$P(X \ge 1) > 0.9$$
I first assumed (this assumption turns out to be incorrect) that $m$ was large enough to allow $X$ to be approximated as
$$X \sim N(0.01m, 0.01m)$$
Then, using the typical standardization approach,
$$P(X\ge 1) > 0.9$$ $$1- P(X <1) > 0.9$$ With continuity correction, we have: $$P(X \le 0.5) < 0.1$$ $$P\left(Z < \frac{0.5 - 0.01m}{\sqrt{0.01m}}\right) < 0.1$$ $$\frac{0.5 - 0.01m}{\sqrt{0.01m}} < -1.29$$
Solving this is trivial and it yields $m \ge 257$.
However, this is incorrect (the answer is $m = 231$). I suppose that it was wrong to assume $m$ was large enough to allow $X$ to be approximated by a normal distribution. The condition required for this is $0.01m > 10$, but clearly this condition does not hold even for the correct answer.
Since $X$ is supported on $\mathbb{N}$ we have $$ P(X\geq 1)>0.9 \,\iff \, P(X=0)<0.1. $$ Thus, you seek the minimal $m$ satisfying $\mathrm{e}^{-0.01m}<0.1$ which exists since $\mathrm{e}^{-0.01m}\to 0$ as $m\to\infty$.