Argument shift Normal Distribution

Question

In a mathematic book I have read following exercise:

We throw a normal coin 10,000 times. The random variable $X$ tells us the number of tails. Give an approximation for $\mathbb{P}(4900 \leq X \leq 5100)$

With the central limit theorem we can calculate this with $\Phi(5100) - \Phi(4900)$ but in the book the solution looks like $\Phi(2) - \Phi(-2)$. Can someone please tell me why we can do this?

Answer 1

In your book $\Phi$ is the cumulative distribution of $Y \sim \mathcal{N}(0,1)$

And $X$ doesn't follow (approximatively) a $\mathcal{N}(0,1)$, but follow (approximatively) a $\mathcal{N}(5000,50^2)$

So

$$P( 4900 \leq X \leq 5100 ) =P( -2 \leq \frac{X-5000}{50} \leq 2 ) $$

And now $\frac{X-5000}{50} \sim \mathcal{N}(0,1)$, so

$$P( 4900 \leq X \leq 5100 ) = \Phi(2)-\Phi(-2)$$