Consider a biased coin, the probability of landing heads up is $\frac{2}{3}$. If this coin is flipped 50 times, then the probability that the number of heads is even is (․).
(A) $25\left(\frac{2}{3}\right)^{50}$
(B) $\frac{1}{2}\left(i-\frac{i}{3^{50}}\right)$
(C) $\frac{1}{2}$
(I) $\frac{3}{2}\left(1+\frac{1}{3^{50}}\right)$
(E) $\frac{\AA}{3}$
(1992, 43rd American High School Mathematics Examination)
The required probability is the sum of the even powers of $\frac{2}{3}$ in the expansion of $\left(\frac{2}{3}+\frac{1}{3}\right)^{50}$. By the above corollary, this is known to be $\frac{1}{2}\left(1+\frac{1}{3^{50}}\right)$.