Do we use normal approximation when discrete distributions are hard to solve?
For example, $P(X\ge 7000)$ where is $X\sim\operatorname{Binomial}(13000, 0.7)$.
Obviously, summing each case manually cannot be done and the summation is difficult to solve (I assume)?
In my textbook, sometimes when something is a dicrete distribution and I calculate it using discrete distribution, I flip to the answer and they used normal approximation to the discrete distribution.
Example:
"Why are you choosing to use normal approximation?!"
As you readily point out, it is a matter of convenience. You can indeed evaluate the binomial distribution $n$ times and the add the results together, but that gets pretty boring really fast. Unless of course you have a computational tool at hand (calculator, computer, $\dots$)
Instead if $n$ is large enough you can use the Normal approximation and the results are strikingly good. The problem is always "what does it mean for $n$ to be large?". And to that there are a few rules you could follow (these are more suggestions that actual rules)
Both $np\gtrsim 5 $ and $n(1-p)\gtrsim 5$
The $|\gamma| < 1/3$ and $n > 5$, with $\gamma$ being the skewness
$$ \gamma = {\frac {1-2p}{\sqrt {np(1-p)}}} $$
I am pretty sure that there are more rules out there, but these have worked out for me in the past