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

Answer 1

The correct option is C

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

Explanation for the correct option:

Step1. Calculate the probability of getting a correct answer :

We know that we have $3$alternative answers but only $1$is correct.

The probability of getting a correct answer $=\frac{1}{3}$

So the probability of getting an incorrect answer is $=1-\frac{1}{3}$

$=\frac{2}{3}$

Step2. Calculate the probability of getting $4$or more correct answers :

The probability of getting $4$or more correct answers

$=$probability of getting $4$correct answers + probability of getting $5$correct answers

$$\begin{gathered} ={}^{5}C_4{\left(\frac{1}{3}\right)}^4{\left(\frac{2}{3}\right)}^{5-4}+{}^{5}C_5{\left(\frac{1}{3}\right)}^5{\left(\frac{2}{3}\right)}^{5-5} \\ =5\times {\left(\frac{1}{3}\right)}^4{\left(\frac{2}{3}\right)}^1+1\times {\left(\frac{1}{3}\right)}^5 \\ =\frac{10}{3^5}+\frac{1}{3^5} \\ =\frac{11}{3^5} \end{gathered}$$

Hence, the correct option is (C).