Chapter 6.2, Problem 25E

Solution

To find: the probability of guessing 30 correct answers on a 30-question multiple choice exam.

The probability of guessing 30 correct answers on a 30-question multiple choice exam is ${\left(0.25\right)}^{30}$.

Given:

A 30-question multiple choice exam has four choices A, B, C, and D for each question.

Concept used:

For a binomial experiment, the probability of $k$ successes in $n$ trails is,

  $P\left(k\text{successes}\right)=C{}_k{p}^k{\left(1-p\right)}^{n-k}$ , where $p$ is the probability of success.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

Calculation:

The probability of guessing a correct choice from four choices is,

  $\begin{array}{c}p=\frac{1}{4}\\ =0.25\end{array}$

Because there are 30 questions in the exam, $n=30$.

Now the probability of guessing 26 correct answers is,

  $\begin{array}{c}P\left(k=30\right)=C{}_{30}{\left(0.25\right)}^{30}{\left(1-0.25\right)}^{30-30}\\ =\frac{30!}{30!\left(30-30\right)!}\times {\left(0.25\right)}^{30}{\left(0.75\right)}^0\\ =\frac{\cancel{30!}}{\cancel{30!}\times 0!}\times {\left(0.25\right)}^{30}\left(1\right)\\ ={\left(0.25\right)}^{30}\end{array}$