Question

A multiple choice examination has $5$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $4$ or more correct answers just by guessing is

• A. $\frac{13}{3^5}$
• B. $\frac{11}{3^5}$
• C. $\frac{10}{3^5}$
• D. $\frac{17}{3^5}$

Answer 1

The correct option is B $\frac{11}{3^5}$

Total number of questions $n=5$
Probability of correct answer $p=\frac{1}{3}$
Probability of incorrect answer $q=\frac{2}{3}$
Required probablility $=P(X\ge 4)$
$=P(X=4)+P(X=5)$

${=}^5{C}_4{\left(\frac{1}{3}\right)}^4{\left(\frac{2}{3}\right)}^1{+}^5{C}_5{\left(\frac{1}{3}\right)}^5{\left(\frac{2}{3}\right)}^0$

$=\frac{10}{3^5}+\frac{1}{3^5}$

$=\frac{11}{3^5}$