Question

On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing ?

Answer 1

The repeated guessing of correct answers from multiple choice questions are Bernoulli trials. Let $X$ represent the number of correct answers by guessing in the set of 5 multiple choice questions.
Probability of getting a correct answer is , $p=\frac{1}{3}$
$\therefore q=1-p=1-\frac{1}{3}=\frac{2}{3}$
Clearly, $X$ has a binomial distribution with $n=5$ and $p=\frac{1}{3}$
$\therefore P(X=x){=}^n{C}_x{q}^{n-x}{p}^x$
${=}^5{C}_x{\left(\frac{2}{3}\right)}^{5-x}\cdot {\left(\frac{1}{3}\right)}^x$
$P\left(guessingmorethan4correctanswers\right)=P(X\ge 4)$
$=P(X=4)+P(X=5)$
${=}^5{C}_4\left(\frac{2}{3}\right)\cdot {\left(\frac{1}{3}\right)}^4{+}^5{C}_5{\left(\frac{1}{3}\right)}^5$
$=5\cdot \frac{2}{3}\cdot \frac{1}{81}+1\cdot \frac{1}{243}$
$=\frac{10}{243}+\frac{1}{243}$
$=\frac{11}{243}$