Chapter 3.2, Problem 25AQP

a.

Summary Introduction

To determine:

The probability of rolling one six-sided die and obtaining 2.

Introduction:

The six-sided die is a cube, on which the numbers from $\text{1}\text{to}\text{6}$ are marked once. When the die is rolled one number is obtained as the outcome.

Explanation of Solution

The number 2 is written only once on the die and can be expressed once. The number of chances that two will appear on die is one. The total numbers that can appear on the die are six.

The probability for obtaining two on a six-sided die is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}\text{2}\text{will}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{1}{6}\end{array}$

b.

Summary Introduction

To determine:

The probability of rolling one six-sided die and obtaining one or two.

Introduction:

The six-sided die is a cube, on which the numbers from $\text{1}\text{to}\text{6}$ are marked once. When the die is rolled one number is obtained as the outcome.

Explanation of Solution

The occurrences of one or two on a die are exclusive events and therefore, by the addition rule both the probabilities can be added to get the total probability of obtaining either one or two.

The number 1 is written only once on the die and can be expressed once. The number of chances that one will appear on die is one. The total numbers that can appear on die are six.

The probability for obtaining one on a six-sided die is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}1\text{will}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{1}{6}\end{array}$

The number 2 is written only once on the die and can be expressed once. The number of chances that two will appear on die is one. The total numbers that can appear on die are six.

The probability for obtaining two on a six-sided die is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}\text{2}\text{will}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{1}{6}\end{array}$

The probability of obtaining one or two is as follows:

$\text{Probability}=\text{Probability}\text{of}\text{obtaining}\text{one}+\text{Probabilty}\text{of}\text{obtaining}\text{two}$

$\begin{array}{c}\text{Probability}=\frac{1}{6}+\frac{1}{6}\\ =\frac{1+1}{6}\\ =\frac{2}{6}\\ =\frac{1}{3}\end{array}$

c.

Summary Introduction

To determine:

The probability of rolling one six-sided die and obtaining an even number.

Introduction:

The six-sided die is a cube, on which the numbers from $\text{1}\text{to}\text{6}$ are marked once. When the die is rolled one number is obtained as the outcome.

Explanation of Solution

There are three even numbers on a die (two, four, and six). The occurrences of two, four, and six on a die are exclusive events and therefore, by the addition rule the probability can be added to get the total probability of obtaining two, four, and six.

The number 2 is written only once on the die and can be expressed once. The number of chances that two will appear on die is one. The total numbers that can appear on die are six.

The probability for obtaining two on a six-sided die is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}2\text{will}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{1}{6}\end{array}$

The number 4 is written only once on the die and can be expressed once. The number of chances that four will appear on die is one. The total numbers that can appear on die are six.

The probability for obtaining four on a six-sided die is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}4\text{will}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{1}{6}\end{array}$

The number 6 is written only once on the die and can be expressed once. The number of chances that six will appear on die is one. The total numbers that can appear on die are six.

The probability for obtaining six on a six-sided die is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}6\text{will}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{1}{6}\end{array}$

The probability of obtaining two, four, and or six is as follows:

$\text{Probability}=\left(\begin{array}{l}\text{Probability}\text{of}\text{obtaining}\text{two}+\\ \text{Probabilty}\text{of}\text{obtaining}\text{four}+\\ \text{Probabilty}\text{of}\text{obtaining}\text{six}\end{array}\right)$

$\begin{array}{c}\text{Probability}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\\ =\frac{1+1+1}{6}\\ =\frac{3}{6}\\ =\frac{1}{2}\end{array}$

d.

Summary Introduction

To determine:

The probability of rolling one six-sided die and obtaining any number but a 6.

Introduction:

The six-sided die is a cube, on which the numbers from $\text{1}\text{to}\text{6}$ are marked once. When the die is rolled one number is obtained as the outcome.

Explanation of Solution

The number 6 is written only once on the die and can be expressed once. The number of chances that six will appear on die is one. The total numbers that can appear on die are six.

The probability for obtaining six on a six-sided die is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}6\text{will}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{1}{6}\end{array}$

The probability of obtaining any number but a six is as follows:

$\begin{array}{c}\text{Probability}=\frac{\text{Number}\text{of}\text{chances}\text{that}6\text{will}\text{not}\text{appear}}{\text{Total}\text{numbers}\text{that}\text{can}\text{appear}\text{on}\text{a}\text{die}}\\ =\frac{5}{6}\end{array}$

Conclusion

The probability of rolling one six-sided die and obtaining 2 is $\frac{1}{6}$. The probability of rolling one six-sided die and obtaining one or two is $\frac{1}{3}$. The probability of rolling one six-sided die and obtaining an even number is $\frac{1}{2}$. The probability of rolling one six-sided die and obtaining any number but a 6 is $\frac{5}{6}$.