Q36 / 90 · JEE Main 2021 (27 Jul Shift 2) · Mathematics

A student appeared in an examination consisting of $8$ true-false type questions. The student guesses the answers with equal probability. The smallest value of $n,$ so that the probability of guessing at least $n$ correct answers is less than $\frac{1}{2},$ is :

Solution

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

$\Rightarrow \sum_{r = n}^{8} {}^{8}C_{r} {(\frac{1}{2})}^{8 - r} {(\frac{1}{2})}^{r} < \frac{1}{2}$

$\Rightarrow \sum_{r = n}^{8} {}^{8}C_{r} {(\frac{1}{2})}^{8} < \frac{1}{2}$

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

$\Rightarrow 256 - ({}^{8}C_{0} + {}^{8}C_{1} + \dots + {}^{8}C_{n - 1}) < 128$

$\Rightarrow {}^{8}C_{0} + {}^{8}C_{1} + \dots + {}^{8}C_{n - 1} > 128$

$\Rightarrow 1 + 8 + 28 + 56 + 70 + {}^{8}C_{5} + \dots + {}^{8}C_{n - 1} > 128$

$\Rightarrow 163 + {}^{8}C_{5} + \dots + {}^{8}C_{n - 1} > 128$

$\Rightarrow n - 1 \geq 4$

$\Rightarrow n \geq 5$

Hence, the correct answer is option: 'A'.