Q. 4.40 - A First Course in Probability

Question

On a multiple-choice exam with $3$possible answers for each of the $5$ questions, what is the probability that a student will get $4$ or more correct answers just by guessing? 

Step-by-Step Solution

Answer

The probability that a student will get $4$or more correct answers are $\frac{11}{243}.$ 

Step 1: Given Information

Given in the question that a multiple-choice exam with $3$ possible answers for each of the $5$ questions. Probability of correct answer $=\frac{1}{3}.$$=\frac{1}{3}.$

Step 2: Solution of the Problem

Probability of correct answer $=\frac{1}{3}$

$P$($4$ correct answers out of $5$)$=\left(\begin{array}{l}5\\ 4\end{array}\right)\times {\left(\frac{1}{3}\right)}^4\times {\left(\frac{2}{3}\right)}^{\text{'}}$

$=\frac{5\times 2}{3^5}$

We get,

$=\frac{10}{243}.$

Step 3: Finding the Probability

$P$($5$correct answers out of $5$)$=\left(\begin{array}{l}5\\ 5\end{array}\right)\times {\left(\frac{1}{3}\right)}^5$

$=\frac{1}{3^5}$

We get,

$=\frac{1}{243}.$

$P$($4$or more correct answers)$=\frac{10}{243}+\frac{1}{243}$

$=\frac{11}{243}.$

Step 4: Final Answer

The probability that a student will get $4$ or more correct answers are $\frac{11}{243}$.