Chapter 13.6, Problem 20E

To determine

To find: the probability of getting at least half answer correct from $10$ question s on a true or false test.

Answer to Problem 20E

The probability of getting at least half answer correct answer out of $10$ questions is $\frac{319}{512}$ .

Explanation of Solution

Given:

The probability of getting the true answer by Maura is $\frac{1}{2}$ and the false answer is also $\frac{1}{2}$ .

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:

The probability of getting the true answer by Maura is $\frac{1}{2}$ and the false answer is also $\frac{1}{2}$ .

  $\begin{array}{l}P\left(\text{true}\right)=\frac{1}{2}.\\ P\left(\text{false}\right)=\frac{1}{2}.\end{array}$

So, the probability of getting at least half correct will be solved through the binomial expression as:

  $\begin{array}{l}P\left(\text{ atleast }5\text{ correct}\right)=P\left(5\text{ correct}\right)+P\left(\text{ atleast }6\text{ correct}\right).\\ \Rightarrow C{}_5\times {\left(\frac{1}{2}\right)}^5\times {\left(\frac{1}{2}\right)}^5+\frac{386}{1024}.\\ \Rightarrow \frac{252}{1024}+\frac{386}{1024}.\\ \Rightarrow \frac{319}{512}.\end{array}$

Hence, the probability of getting $5$ correct answer out of $10$ questions is $\frac{319}{512}$ .