Question

A quiz consists of 10 multiple-choice questions, each with 6 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 50 %.

Answer 1

Solution: The probability of passing if the minimum passing grade is 50% is 0.0155

Explanation:

There are total 10 questions $(n=10)$ .

The probability of getting a correct answer is $\frac{1}{6}$, $p=\frac{1}{6}$

The minimum passing grade is 50%, therefore, the minimum number of questions required to pass the grade is:

$0.5 \times 10=5$

Therefore we have to find $P(x\geq 5)$

Now using the binomial probability model, we have:

$P(x\geq 5)=\binom{10}{5}\left ( \frac{1}{6} \right )^{5}\left ( \frac{5}{6} \right )^{10-5}+\binom{10}{6}\left ( \frac{1}{6} \right )^{6}\left ( \frac{5}{6} \right )^{10-6}+\binom{10}{7}\left ( \frac{1}{6} \right )^{7}\left ( \frac{5}{6} \right )^{10-7}+\binom{10}{8}\left ( \frac{1}{6} \right )^{8}\left ( \frac{5}{6} \right )^{10-8}+\binom{10}{9}\left ( \frac{1}{6} \right )^{9}\left ( \frac{5}{6} \right )^{10-9}+\binom{10}{10}\left ( \frac{1}{6} \right )^{10}\left ( \frac{5}{6} \right )^{10-10}$

  $\small =0.01302381+0.00217064+0.00024807+0.00001861+0.00000083+0.00000002$

$\small =0.0155$

Therefore, there is 0.0155 probability of passing if the minimum passing grade is 50%.