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)

$\frac{7}{128}$

Answer 2

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