2010 AMC 12A Problem 15

A coin is altered so that the probability that it lands on heads is less than \frac{1}{2} and when the coin is flipped four times, the probability of an equal number of heads and tails is \frac{1}{6}. What is the probability that the coin lands on heads?

\textbf{(A)}\ \frac{\sqrt{15}-3}{6} \qquad \textbf{(B)}\ \frac{6-\sqrt{6\sqrt{6}+2}}{12} \qquad \textbf{(C)}\ \frac{\sqrt{2}-1}{2} \qquad \textbf{(D)}\ \frac{3-\sqrt{3}}{6} \qquad \textbf{(E)}\ \frac{\sqrt{3}-1}{2}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.