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

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 $\mathrm{n}$, so that the probability of guessing at least ' $n$ ' correct answers is less than $\frac{1}{2}$, is :

1. 5

2. 6

3. 3

4. 4

Correct Option: 1

Solution:

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

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

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

$\Rightarrow{ }^{8} \mathrm{C}_{\mathrm{n}}+{ }^{8} \mathrm{C}_{\mathrm{n}+1}+\ldots .+{ }^{8} \mathrm{C}_{8}<128$

$\Rightarrow 256-\left({ }^{8} \mathrm{C}_{0}+{ }^{8} \mathrm{C}_{1}+\ldots .+{ }^{8} \mathrm{C}_{\mathrm{n}-1}\right)<128$

$\Rightarrow{ }^{8} \mathrm{C}_{0}+{ }^{8} \mathrm{C}_{1}+\ldots .+{ }^{8} \mathrm{C}_{\mathrm{n}-1}>128$

$\Rightarrow \mathrm{n}-1 \geq 4$

$\Rightarrow \mathrm{n} \geq 5$