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, $P\left(X \ge 8\right)$
$= P\left(X = 8\right) + P\left(X = 9\right) + P\left(X = 10\right)$
$= \,{}^{10}C_{8}\left(\frac{1}{2}\right)^{8}\left(\frac{1}{2}\right)^{2}+\,{}^{10}C_{9}\left(\frac{1}{2}\right)^{9}\left(\frac{1}{2}\right)+\,{}^{10}C_{10}\left(\frac{1}{2}\right)^{10}$
$= \left(\frac{1}{2}\right)^{10}\left[\,{}^{10}C_{8}+\,{}^{10}C_{9}+\,{}^{10}C_{10}\right]$
$= \left(\frac{1}{2}\right)^{10} \times\left[45+10+1\right]$
$= 56\times\left(\frac{1}{2}\right)^{10} = \frac{7}{128}$