A student appeared in an examination consisting of 8 true-false...

$P\left(E\right) < \dfrac{1}{2}$

$⇒ \mathop{∑}_{r = n}^{8} Cr 8 {\left(\dfrac{1}{2}\right)}^{8 - r}{\left(\dfrac{1}{2}\right)}^{r} < \dfrac{1}{2}$

$⇒ \mathop{∑}_{r = n}^{8} Cr 8 {\left(\dfrac{1}{2}\right)}^{8} < \dfrac{1}{2}$

$⇒ Cn 8 + Cn + 1 8 + \ldots + C8 8 < 128$

$⇒ 256 - \left(C0 8 + C1 8 + \ldots + Cn - 1 8\right) < 128$

$⇒ C0 8 + C1 8 + \ldots + Cn - 1 8 > 128$

$⇒ 1 + 8 + 28 + 56 + 70 + C5 8 + \ldots + Cn - 1 8 > 128$

$⇒ 163 + C5 8 + \ldots + Cn - 1 8 > 128$

$⇒ n - 1 \ge 4$

$⇒ n \ge 5$