Chapter 13.6, Problem 3CFU

To determine

To find: the probability of getting exactly $2$ correct answers out of $5$ true and false question of quiz.

Answer to Problem 3CFU

The probability of getting exactly $2$ correct answer is $\frac{15}{8}$ .

Explanation of Solution

Given:

The true/false quiz has $5$ questions.

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 true/false quiz has $5$ questions.

Since the choice is true and false so the probability will be:

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

So, the probability of getting exactly $2$ correct answer is:

  $\begin{array}{l}P\left(\text{exactly }2\text{ correct answers}\right):\\ \Rightarrow C{}_2\times {\left(\frac{1}{2}\right)}^2\times {\left(\frac{1}{2}\right)}^3.\\ \Rightarrow 60\times \frac{1}{4}\times \frac{1}{8}=\frac{15}{8}.\end{array}$

Hence, the probability of getting exactly $2$ correct answer is $\frac{15}{8}$ .