Question

The probability of guessing correctly at least $8$ out of $10$ answers on a true-false type examination is

• A. $\frac{7}{64}$
• B. $\frac{7}{128}$
• C. $\frac{45}{1024}$
• D. $\frac{7}{41}$

Answer 1

Answer: (B)

It is a binomial distribution case with $n=10$ and probability of guessing correctly, $p=\frac{1}{2}$
$\Rightarrow q=1-p=1-\frac{1}{2}=\frac{1}{2}$
$\therefore$ Required probability, $P\left(X\ge 8\right)$
$=P\left(X=8\right)+P\left(X=9\right)+P\left(X=10\right)$
$={}^{10}{C}_8{\left(\frac{1}{2}\right)}^8{\left(\frac{1}{2}\right)}^2+{}^{10}{C}_9{\left(\frac{1}{2}\right)}^9\left(\frac{1}{2}\right)+{}^{10}{C}_{10}{\left(\frac{1}{2}\right)}^{10}$
$={\left(\frac{1}{2}\right)}^{10}\left[{}^{10}{C}_8+{}^{10}{C}_9+{}^{10}{C}_{10}\right]$
$={\left(\frac{1}{2}\right)}^{10}\times \left[45+10+1\right]$
$=56\times {\left(\frac{1}{2}\right)}^{10}=\frac{7}{128}$