Chapter 5.1, Problem 29E

True-false exam: A section of an exam contains four true-false questions. A completed exam is selected at random, and the four answers are recorded.

1. List all 16 outcomes in the sample space.
2. Assuming the outcomes to be equally find the probability that all the answers are the same.
3. Assuming the outcomes to be equally find the probability that exactly one of the four answers is "True."
4. Assuming the outcomes to be equally likely, find the probability that two of the answers are "True" and two of the answers are "false."

To determine

To list all the outcomes of the sample space.

Answer to Problem 29E

  $\left\{\begin{array}{l}\text{TTTT}\text{,}\text{TTTF}\text{,}\text{FTTT}\text{,}\text{TFTT}\text{,}\text{TTFT}\text{,}\text{TTFF}\text{,}\text{TFTF}\text{,}\text{FTFT}\text{,}\text{TFFT}\text{,}\text{FTTF}\text{,}\text{FFTT}\text{,}\text{TFFF}\text{,}\text{FTFF}\text{,}\text{FFTF}\text{,}\\ \text{FFFT}\text{,}\text{FFFF}\end{array}\right\}$

Explanation of Solution

Given:

In an exam, there is a section that consists of 4 true/false questions.

Calculation:

The total questions are four and there are two choices for each question, which is true or false.

Therefore, total outcomes are $2\times 2\times 2\times 2=16$ .

Let T represents true and F represents false.

The sample space is,

  $\left\{\begin{array}{l}\text{TTTT}\text{,}\text{TTTF}\text{,}\text{FTTT}\text{,}\text{TFTT}\text{,}\text{TTFT}\text{,}\text{TTFF}\text{,}\text{TFTF}\text{,}\text{FTFT}\text{,}\text{TFFT}\text{,}\text{FTTF}\text{,}\text{FFTT}\text{,}\text{TFFF}\text{,}\text{FTFF}\text{,}\text{FFTF}\text{,}\\ \text{FFFT}\text{,}\text{FFFF}\end{array}\right\}$

To determine

To find the probability that all the answers are the same.

Answer to Problem 29E

The probability that all the answers are the same is $\frac{1}{8}$ .

Explanation of Solution

Calculation:

From subpart (a), we have the sample space

  $\left\{\begin{array}{l}\text{TTTT}\text{,}\text{TTTF}\text{,}\text{FTTT}\text{,}\text{TFTT}\text{,}\text{TTFT}\text{,}\text{TTFF}\text{,}\text{TFTF}\text{,}\text{FTFT}\text{,}\text{TFFT}\text{,}\text{FTTF}\text{,}\text{FFTT}\text{,}\text{TFFF}\text{,}\text{FTFF}\text{,}\text{FFTF}\text{,}\\ \text{FFFT}\text{,}\text{FFFF}\end{array}\right\}$

Let $A$ be the event that all the answers are same.

Therefore $A=\left\{\text{TTTT}\text{,}\text{FFFF}\right\}$ .

Hence the probability that all the answers are the same is,

  $\begin{array}{l}P\left(A\right)=\frac{\text{Number}\text{of}\text{outcomes}\text{in}A}{\text{Number}\text{of}\text{outcomes}\text{in}\text{the}\text{sample}\text{space}}\\ =\frac{2}{16}\\ =\frac{1}{8}\end{array}$

To determine

To find the probability that exactly one of the four answers is true.

Answer to Problem 29E

The probability that exactly one of the four answers is true is $\frac{1}{4}$ .

Explanation of Solution

Calculation:

From subpart (a), we have the sample space

  $\left\{\begin{array}{l}\text{TTTT}\text{,}\text{TTTF}\text{,}\text{FTTT}\text{,}\text{TFTT}\text{,}\text{TTFT}\text{,}\text{TTFF}\text{,}\text{TFTF}\text{,}\text{FTFT}\text{,}\text{TFFT}\text{,}\text{FTTF}\text{,}\text{FFTT}\text{,}\text{TFFF}\text{,}\text{FTFF}\text{,}\text{FFTF}\text{,}\\ \text{FFFT}\text{,}\text{FFFF}\end{array}\right\}$

Let $B$ be the event that exactly one of the four answers are true.

Therefore $B=\left\{\text{TFFF}\text{,}\text{FTFF}\text{,}\text{FFTF}\text{,}\text{FFFT}\right\}$ .

Hence the probability that exactly one of the four answers is true is,

  $\begin{array}{l}P\left(B\right)=\frac{\text{Number}\text{of}\text{outcomes}\text{in}B}{\text{Number}\text{of}\text{outcomes}\text{in}\text{the}\text{sample}\text{space}}\\ =\frac{4}{16}\\ =\frac{1}{4}\end{array}$

To determine

To find the probability that two of the answers are true and two of the answers are false.

Answer to Problem 29E

The probability that two of the answers are true and two of the answers are false is $\frac{3}{8}$ .

Explanation of Solution

Calculation:

From subpart (a), we have the sample space

  $\left\{\begin{array}{l}\text{TTTT}\text{,}\text{TTTF}\text{,}\text{FTTT}\text{,}\text{TFTT}\text{,}\text{TTFT}\text{,}\text{TTFF}\text{,}\text{TFTF}\text{,}\text{FTFT}\text{,}\text{TFFT}\text{,}\text{FTTF}\text{,}\text{FFTT}\text{,}\text{TFFF}\text{,}\text{FTFF}\text{,}\text{FFTF}\text{,}\\ \text{FFFT}\text{,}\text{FFFF}\end{array}\right\}$

Let $C$ be the event that two of the answers are true and two of the answers are false.

Therefore $C=\left\{\text{TTFF}\text{,}\text{TFTF}\text{,}\text{FTFT}\text{,}\text{TFFT}\text{,}\text{FTTF}\text{,}\text{FFTT}\right\}$ .

Hence the probability that two of the answers are true and two of the answers are false is,

  $\begin{array}{l}P\left(C\right)=\frac{\text{Number}\text{of}\text{outcomes}\text{in}C}{\text{Number}\text{of}\text{outcomes}\text{in}\text{the}\text{sample}\text{space}}\\ =\frac{6}{16}\\ =\frac{3}{8}\end{array}$