Chapter 13.6, Problem 25E

To determine

To find: the probability of getting half answer correct from the $10$ multiple choice questions.

Answer to Problem 25E

The probability of getting half or $5$ answers correct is $0.058$ .

Explanation of Solution

Given:

The correct choice is $1$ out of $4$ option in multiple choice question.

Concept used:

Binomial probability refers to the probability of exactly $x$ successes on $n$ repeated trials in an Experiment which has two possible outcomes common called the binomial experiment.

If the probability of success on an individual trial is $p$ and $q$ where $p+q=1$ or $q=1-p$ .

Then the binomial probability is:

  $C{}_x{p}^x{q}^{n-x}\text{ or }C{}_x{p}^x{\left(1-p\right)}^{n-x}$ .

The probability is minimum $0$ and maximum $1$ .

  $0\le p\left(A\right)\le 1.$

If $p\left(A\right)+p\left(B\right)=1$ then $p\left(B\right)=1-p\left(A\right)$ .

At least $x$ in the probability means considering the $x$ and greater than $x$ .

At most $x$ in the probability means considering the $x$ and lower than $x$ .

Calculation:

According to the given correct choice is $1$ out of $4$ option in multiple choice question.

  $\begin{array}{l}P\left(\text{correct}\right)=\frac{1}{4}.\\ P\left(\text{not correct}\right)=1-\frac{1}{4}=\frac{3}{4}.\end{array}$

Kyle guesses all $10$ questions.

The probability of getting half answers correct mean out of $10$ question answers of $5$ questions is correct so, the probability of getting $5$ answers corrects can be solved by binomial expression as:

  $\begin{array}{l}P\left(5\text{ correct answers}\right)=C{}_5\times {\left(\frac{1}{4}\right)}^5\times {\left(\frac{3}{4}\right)}^5.\\ \Rightarrow 252\times \frac{1}{1024}\times \frac{243}{1024}.\\ \Rightarrow \mathrm{0.058.}\end{array}$

Hence, the probability of getting half or $5$ answers correct is $0.058$ .