Trying to prove $\frac{1}{2^{2n+1}}\sum_{r=0}^{n}\binom{2n-r}{n}2^r=\frac 12$

Question

In order to solve a probability problem I had to prove the following equality:

$$\frac{1}{2^{2n+1}}\sum_{r=0}^{n}\binom{2n-r}{n}2^r=\frac 12$$

I tried to prove it but to no avail. The main problem is that I've never studied combinatorics, and our introduction to the binomial coefficient was very minimal. The only two things I know about them is their symmetry, and the binomial theorem of Newton. However, In order to use one of these properties or both, I need the summation variable $r$ to be on the bottom of the coefficient; rather than on the top of it.

Thanks!