Question

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

Answer 1

Let X denotes the number of correct answers given by the candidate. It is a case of Bernoulli trails with n=5, where success is guessing a correct answer to a multiple choice question with 3 options.

$\therefore$ p=$\frac{1}{3}andq=1-p=1-\frac{1}{3}andq=\frac{2}{3},$

Clearly, X has a binomial distribution with n=5, p=$\frac{1}{3}andq=\frac{2}{3}$

P(X=r) = ${}^5{C}_r{\left(\frac{1}{3}\right)}^r.{\left(\frac{2}{3}\right)}^{5-r}$

Required probablity + P (four or more correct answers)

= $P(X\ge 4)=P\left(4\right)+P\left(5\right)={}^5{C}_4{p}^{4}q+{}^5{C}_5{p}^5{q}^0={}^5{C}_4{\left(\frac{1}{3}\right)}^4\frac{2}{3}+{}^5{C}_5{\left(\frac{1}{3}\right)}^5.{\left(\frac{2}{3}\right)}^0=5\times \frac{2}{3}\times \frac{1}{3^4}=\frac{11}{3^5}=\frac{11}{243}$