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Roll: Definition and Example | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">RollRoll: Definition and ExampleTable of ContentsDefinition of Roll

In mathematics, a roll refers to the act of generating a random outcome using objects that have multiple equally possible results, most commonly with dice. Rolling is a fundamental concept in probability and statistics, where we study the likelihood of different outcomes. When we roll a standard six-sided die, each face (numbered 111 through 666) has an equal chance of landing face up, giving each number a probability of 16\frac{1}{6}61​ or about 16.7%16.7\%16.7%.

Rolling also has geometric meaning in mathematics. In this context, rolling describes the motion of a curve or surface as it moves along another curve or surface while maintaining contact, without slipping or sliding. This type of movement combines rotation and translation, and it appears in various applications from wheel mechanics to cycloid curves. Understanding rolling motion helps us solve problems involving distance, rotation, and the path traced by points on rolling objects.

Examples of Roll Example 1: Finding Probability of Dice Rolls Problem:

When rolling a standard six-sided die once, what is the probability of rolling an even number?

Step-by-step solution:

Step 1, Identify all possible outcomes when rolling a six-sided die.

The possible outcomes are 111, 222, 333, 444, 555, and 666.

Step 2, Count the total number of possible outcomes.

There are 666 possible outcomes.

Step 3, Identify the outcomes that match what we're looking for (even numbers).

Even numbers on the die are 222, 444, and 666.

Step 4, Count the number of favorable outcomes.

There are 333 even numbers.

Step 5, Calculate the probability using the formula:

Probability=Number of favorable outcomesTotal number of possible outcomes\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}Probability=Total number of possible outcomesNumber of favorable outcomes​ Probability=36=12=50%\text{Probability} = \frac{3}{6} = \frac{1}{2} = 50\%Probability=63​=21​=50%

Step 6, Therefore, the probability of rolling an even number is 12\frac{1}{2}21​ or 50%50\%50%.

Example 2: Finding the Sum of Two Dice Rolls Problem:

When rolling two standard six-sided dice, what is the probability of rolling a sum of 777?

Step-by-step solution:

Step 1, Determine all possible outcomes when rolling two dice.

Each die has 666 possible outcomes, so there are 6×6=366 \times 6 = 366×6=36 total possible combinations.

Step 2, List the possible ways to get a sum of 7:

(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)

Step 3, Count the number of favorable outcomes.

There are 666 ways to roll a sum of 777.

Step 4, Calculate the probability:

Probability of rolling a sum of 7=636=16≈16.7%\text{Probability of rolling a sum of 7} = \frac{6}{36} = \frac{1}{6} ≈ 16.7\%Probability of rolling a sum of 7=366​=61​≈16.7%

Step 5, Therefore, the probability of rolling a sum of 777 with two dice is 16\frac{1}{6}61​ or approximately 16.7%.

Example 3: Finding the Distance a Circle Rolls Problem:

A circle with a radius of 444 inches rolls along a straight line without slipping. How far does the center of the circle move when the circle makes exactly one complete rotation?

Step-by-step solution:

Step 1, Recall that when a circle rolls without slipping, the distance its center moves equals the distance around the circle (its circumference).

Step 2, Find the circumference of the circle using the formula:

Circumference=2πr\text{Circumference} = 2\pi rCircumference=2πr, where rrr is the radius of the circle.

Step 3, Substitute the given radius (444 inches) into the formula.

Circumference=2π×4 inches\text{Circumference} = 2\pi \times 4 \text{ inches}Circumference=2π×4 inches Circumference=8π inches\text{Circumference} = 8\pi \text{ inches}Circumference=8π inches

Step 4, Calculate the approximate value using π≈3.14\pi \approx 3.14π≈3.14.

Circumference≈8×3.14 inches≈25.12 inches\text{Circumference} \approx 8 \times 3.14 \text{ inches} \approx 25.12 \text{ inches}Circumference≈8×3.14 inches≈25.12 inches

Step 5, Therefore, when the circle makes one complete rotation, its center moves approximately 25.1225.1225.12 inches, or exactly 8π8\pi8π inches.

Comments(1)TTechGeekIvyNovember 5, 2025I've used this 'roll' def to explain probability to my students. It's clear & the examples really help them grasp the concept!

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