At least $5$ events with a probability of at least $0.9$

Let the observation of a drone from a certain earth station is possible with a probability of $0.1$. How many times must the drone fly over the station for the operator to make at least $5$ observations with a probability of at least $0.9975$?

I calculated the exact value on the computer using the Bernoulli formula, estimating the probability that the number of successes is no more than $4$. It turned out $n=131$. It's right?

Basically the inequality is $1-P(X\leq 4)\geq 0.9975$, where $X\sim Bin(n,0.1)$.

$$\begin{eqnarray*} &\texttt{Adding }P(X\leq 4) \\ 1\geq 0.9975 +P(X\leq 4)\\ & \texttt{Subtracting }0.9975\\ 1-0.9975\geq P(X\leq 4) \\ 0.0025\geq P(X\leq 4) \end{eqnarray*}$$

Then we can calculate $P(X\leq 4)$ for $n=130$ and $n=131$

$$P(X\leq 4|n=130)=\sum_{x=0}^{4}\binom{130}{x}\cdot 0.1^x\cdot 0.9^{130-x}=0.002635...>0.0025\ \ \times$$

$$P(X\leq 4|n=131)=\sum_{x=0}^{4}\binom{131}{x}\cdot 0.1^x\cdot 0.9^{131-x}=0.00244...<0.0025 \ \ \checkmark$$

So n=131 seems enough. The poisson distribution is an approximation only. But it is not so bad in this case.