On a multiple choice examination with four possible answers (out of which only one is correct) for each of five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{1}{64} \)
The five guesses are independent Bernoulli trials, so the number of correct answers X is binomial with n = 5, \( p=\frac{1}{4} \), \( q=\frac{3}{4} \). “Four or more” means X = 4 or X = 5.
\( \displaystyle P(X=4)={}^{5}C_{4}\left(\frac{1}{4}\right)^4\left(\frac{3}{4}\right) \)
\( \displaystyle =5\cdot\frac{1}{256}\cdot\frac{3}{4} \)
\( \displaystyle =\frac{15}{1024} \)
\( \displaystyle P(X=5)=\left(\frac{1}{4}\right)^5=\frac{1}{1024} \)
\( \displaystyle P(X\ge 4)=\frac{15+1}{1024} \)
\( \displaystyle =\frac{16}{1024}=\frac{1}{64} \)