Answer: D

Small Hint:

Track the difference between the probabilities of an even and an odd number of heads

Big Hint:

One toss multiplies that difference by $P(T)-P(H)$

Solution:

Let $E_n$ and $O_n$ be the probabilities of an even and odd number of heads after $n$ tosses. Then $E_n+O_n=1,$ while $$\begin{aligned} E_{n+1}-O_{n+1} &=\left(\frac13-\frac23\right)\\ &\qquad\cdot(E_n-O_n). \end{aligned}$$ Since $E_0-O_0=1,$ we have $E_{50}-O_{50}=(-\frac{1}{3})^{50}=3^{-50}.$ Solving the sum and difference equations gives $$E_{50}=\frac12\left(1+\frac1{3^{50}}\right).$$

Thus the correct answer is D.