Let $F\left( n \right) = \sum\limits_{k = 0}^n {{{\left( {C_n^k{p^k}{{\left( {1 - p} \right)}^{n - k}}} \right)}^2}} $, prove $F\left( n \right) \ge F\left( {n + 1} \right)$.

UPDATE: More general, denote $F\left( n \right) = \sum\limits_{k = 0}^n {C_n^kp_1^kq_1^{n - k}C_n^kp_2^kq_2^{n - k}}$, where ${q_1} = 1 - {p_1}$ and ${q_2} = 1 - {p_2}$, prove $F\left( n \right) \ge F\left( {n + 1} \right)$.