jmc

There are $2^{10}=1024$ possible outcomes of the 10 coin flips. The probability that we flip at least 6 heads is equal to the probability that we flip at least 6 tails, by symmetry. Let's call this probability $p$. The only other possibility is that we flip exactly 5 heads and 5 tails, for which the probability is $\frac{\left(\genfrac{}{}{0ex}{}{10}{5}\right)}{2^{10}}=\frac{252}{1024}=\frac{63}{256}$. Therefore, $\frac{63}{256}+2p=1$, giving $$p=\frac{1}{2}\left(1-\frac{63}{256}\right)=\boxed{\frac{193}{512}}$$