Question

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:

• A. 5
• B. 6
• C. 3
• D. 4

Answer 1

Answer: (A)

$P(E)<\frac{1}{2}$
$\Rightarrow \displaystyle\sum_{r=n}^{8}{ }^{8} C_{r}\left(\frac{1}{2}\right)^{8-r}\left(\frac{1}{2}\right)^{r}< \frac{1}{2}$
$\Rightarrow \displaystyle\sum_{r=n}^{8}{ }^{8} C_{r}\left(\frac{1}{2}\right)^{8}<\frac{1}{2}$
$\Rightarrow{ }^{8} C_{n}+{ }^{8} C_{n+1}+\ldots \ldots+{ }^{8} C_{8}<128$
$\Rightarrow 256-\left({ }^{8} C_{0}+{ }^{8} C_{1}+\ldots \ldots+{ }^{8} C_{n-1}\right)<128$
$\Rightarrow{ }^{8} C_{0}+{ }^{8} C_{1}+\ldots .++{ }^{8} C_{n-1}>128$
$\Rightarrow n-1 \geq 4$
$\Rightarrow n \geq 5$