Use central limit to solve a problem

Question

I am asked to solve the following question using central limit theorem.
In an election between two candidates, A and B, one million individuals cast theirvote. Among these, 2000 know candidate A from her election campaign and vote unanimously for her.The remaining 998000 voters are undecided and make their decision independently of each other byflipping a fair coin.Approximate the probability pA that candidate A wins up to 3 significant figures.
It's easy to solve directly. PA = $\frac{0.5(10000-2000)+2000}{10000}$ = 0.501. However, I am quite confused about how to solve this problem by central limit theorem.

Answer 1

Let $X=$ number of voters for candidate A among the remaining $998,000$ individuals. Then $X$ is binomial with $n=998000$ and $p=1/2$, therefore approximately normal with $\mu=np=499000$ and $\sigma^2np(1-p)=249500$. Now calculate $P(X>498000)$, which is surprisingly high.