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)

$P(E)<\frac{1}{2}$
$\Rightarrow \sum _{r=n}^8{}^8{C}_r{\left(\frac{1}{2}\right)}^{8-r}{\left(\frac{1}{2}\right)}^r<\frac{1}{2}$
$\Rightarrow \sum _{r=n}^8{}^8{C}_r{\left(\frac{1}{2}\right)}^8<\frac{1}{2}$
$\Rightarrow {}^8{C}_n+{}^8{C}_{n+1}+\dots \dots +{}^8{C}_8<128$
$\Rightarrow 256-\left({}^8{C}_0+{}^8{C}_1+\dots \dots +{}^8{C}_{n-1}\right)<128$
$\Rightarrow {}^8{C}_0+{}^8{C}_1+\dots .++{}^8{C}_{n-1}>128$
$\Rightarrow n-1\ge 4$
$\Rightarrow n\ge 5$