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. $4$
• B. $3$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

$P(x=n)={}^8{C}_n{\left(\frac{1}{2}\right)}^n{\left(\frac{1}{2}\right)}^{8-n}={}^8{C}_n{\left(\frac{1}{2}\right)}^8$,
$P(x=n+1)={}^8{C}_{n+1}{\left(\frac{1}{2}\right)}^8$
and
$P(x=8)={}^8{C}_8{\left(\frac{1}{2}\right)}^8$
According to question
$P(x=n)+P(x=n+1)+\cdots +P(x=8)<\frac{1}{2}$
$\Rightarrow {\left(\frac{1}{2}\right)}^8({}^8{C}_n{+}^8{C}_{n+1}+\cdots {+}^8{C}_8)<\frac{1}{2}$
$\Rightarrow 2^8-{(}^8{C}_0{+}^8{C}_1+\cdots {+}^8{C}_{n-1})<2^7$
$\Rightarrow {}^8{C}_0{+}^8{C}_1+\cdots {+}^8{C}_{n-1}>2^7$
Minimum value of $n-1=4\Rightarrow n=5$