A student appeared in an examination consisting of 8 true-false type questions. The student guesses the…

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

$\Rightarrow \sum _{r=n}^8{}^{8}C_{\mathrm{r}}{\left(\frac{1}{2}\right)}^{8-\mathrm{r}}{\left(\frac{1}{2}\right)}^{\mathrm{r}}<\frac{1}{2}$

$\Rightarrow \sum _{r=n}^8{}^{8}C_r{\left(\frac{1}{2}\right)}^8<\frac{1}{2}$

$\Rightarrow {}^{8}C_n+{}^{8}C_{n+1}+\dots +{}^{8}C_8<128$

$\Rightarrow 256-\left({}^{8}C_0+{}^{8}C_1+\dots +{}^{8}C_{n-1}\right)<128$

$\Rightarrow {}^{8}C_0+{}^{8}C_1+\dots +{}^{8}C_{n-1}>128$

$\Rightarrow 1+8+28+56+70+{}^{8}C_5+\dots +{}^{8}C_{n-1}>128$

$\Rightarrow 163+{}^{8}C_5+\dots +{}^{8}C_{n-1}>128$

$\Rightarrow n-1\ge 4$

$\Rightarrow n\ge 5$