Question

Sol. [Key Concept: By symmetry $P(more\ even\ than\ odd) = \frac{1}{2}(1 - P(equal))$.]

Probability of getting more even than odd $=$ probability of getting less even than odd outcomes

$$= \frac{1}{2}(1 - \text{~}\text{probability of getting equal odd}\text{~}\&\text{~}\text{even outcomes}\text{~})$$

$$\begin{matrix} & = \frac{1}{2}\left( 1 - \frac{\ ^{8}C_{0}^{2} + \ ^{8}C_{1}^{2} + \ ^{8}C_{2}^{2}\ldots\ldots + \ ^{8}C_{8}^{2}}{2^{16}} \right) \\ & \ = \frac{1}{2}\left( 1 - \frac{\ ^{16}C_{8}}{2^{16}} \right) \end{matrix}$$