Chapter 11.4, Problem 47STP

a.

To determine

What is the probability that she will get, $7$ questions correct?

Answer to Problem 47STP

  $\boxed{0.3\%}$

Explanation of Solution

Given information:

Carly is taking a $10$ question multiple-choices test, in which each question has four choices. If she guesses on each question, what is the probability that she will get, $7$ questions correct?

Calculation:

Consider Carly is taking a $10$ question multiple-choices test, in which each question has four choices

Suppose she guesses on each question.

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $p$ represents the probability of success and $q$ represents the probability of failure.

Well $1$ out of $4$ choices is correct, the probability of success is,

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

The probability of failure is,

  $\begin{array}{l}q=1-p\\ =1-0.25\\ =0.75\end{array}$

Now put calculated values in expression,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $\begin{array}{l}P\left(7\right)=C{}_7{\left(0.25\right)}^7{\left(0.75\right)}^{10-7}\\ =0.003\\ =0.3\%\end{array}$

Hence, the probability that she will get, $7$ Questions correct is $\boxed{0.3\%}$

b.

To determine

What is the probability that she will get, $9$ questions correct?

Answer to Problem 47STP

  $\boxed{0.003\%}$

Explanation of Solution

Given information:

Carly is taking a $10$ question multiple-choices test, in which each question has four choices. If she guesses on each question, what is the probability that she will get, $9$ questions correct?

Calculation:

Consider Carly is taking a $10$ question multiple-choices test, in which each question has four choices

Suppose she guesses on each question.

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $p$ represents the probability of success and $q$ represents the probability of failure.

Well $1$ out of $4$ choices is correct, the probability of success is,

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

The probability of failure is,

  $\begin{array}{l}q=1-p\\ =1-0.25\\ =0.75\end{array}$

Now put calculated values in expression,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $\begin{array}{l}P\left(9\right)=C{}_9{\left(0.25\right)}^9{\left(0.75\right)}^{10-9}\\ =0.00003\\ =\boxed{0.003\%}\end{array}$

Hence, the probability that she will get, $9$ questions correct is $\boxed{0.003\%}$

c.

To determine

What is the probability that she will get, zero questions correct?

Answer to Problem 47STP

  $\boxed{5.6\%}$

Explanation of Solution

Given information:

Carly is taking a $10$ question multiple-choices test, in which each question has four choices. If she guesses on each question, what is the probability that she will get, zero questions correct?

Calculation:

Consider Carly is taking a $10$ question multiple-choices test, in which each question has four choices

Suppose she guesses on each question.

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $p$ represents the probability of success and $q$ represents the probability of failure.

Well $1$ out of $4$ choices is correct, the probability of success is,

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

The probability of failure is,

  $\begin{array}{l}q=1-p\\ =1-0.25\\ =0.75\end{array}$

Now put calculated values in expression,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $\begin{array}{l}P\left(0\right)=C{}_0{\left(0.25\right)}^0{\left(0.75\right)}^{10-0}\\ =0.056\\ =\boxed{5.6\%}\end{array}$

Hence, the probability that she will get, zero questions correct is $\boxed{5.6\%}$

d.

To determine

What is the probability that she will get, $3$ questions correct?

Answer to Problem 47STP

  $\boxed{25.0\%}$

Explanation of Solution

Given information:

Carly is taking a $10$ question multiple-choices test, in which each question has four choices. If she guesses on each question, what is the probability that she will get, $3$ questions correct?

Calculation:

Consider Carly is taking a $10$ question multiple-choices test, in which each question has four choices

Suppose she guesses on each question.

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $p$ represents the probability of success and $q$ represents the probability of failure.

Well $1$ out of $4$ choices is correct, the probability of success is,

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

The probability of failure is,

  $\begin{array}{l}q=1-p\\ =1-0.25\\ =0.75\end{array}$

Now put calculated values in expression,

  $P\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

  $\begin{array}{l}P\left(3\right)=C{}_3{\left(0.25\right)}^3{\left(0.75\right)}^{10-3}\\ =0.250\\ =\boxed{25.0\%}\end{array}$

Hence, the probability that she will get, $3$ questions correct is $\boxed{25.0\%}$