The numbers in question here, of course, can be computed exactly. For example, using bignum or GAP (or even WolframAlpha -- the exactly link won't work on here, but I'm sure you can type in "(100 choose 50)*2^(-100)" yourself).
On my home computer, in GAP, it took less than a millisecond. To make it more interesting, I also computed ${100000 \choose 50000} \cdot 2^{-100000}$, which took a bit more than 17 seconds.
gap> Binomial(100,50)/2^100;
12611418068195524166851562157/158456325028528675187087900672
gap> time;
0
gap> Binomial(100000,50000)/2^100000;
<>/<>
gap> time;
17266
In fact, provided the coin has probability 1/2, the probability will always have a terminating decimal expansion (since binomial coefficients are integers, and 2 divides 10). Here it is in this case:
0.0795892373871787614981270502421704614029315404247333213573478705171737601631321012973785400390625