Question

Three dice are tossed. How many simple events are in the sample space?

Short answer

Answer: There are 216 simple events in the sample space when three dice are tossed.

Step-by-step solution

Determine the number of outcomes for each die

A single die has 6 faces, each with a unique number from 1 to 6. Therefore, each die can have 6 different outcomes for each toss.

Determine the total number of outcomes for tossing three dice

Since each die has 6 possible outcomes and there are 3 dice, we can find the total number of outcomes for all three dice by calculating the Cartesian product of the outcomes of each die. In other words, we can find all the possible combinations of outcomes by multiplying the number of outcomes for each die. So, the total number of outcomes for three dice can be found using the formula: Total outcomes = 6 (for first die) * 6 (for second die) * 6 (for third die)

Calculate the total number of outcomes for all three dice

Using the formula from Step 2, we can calculate the total number of outcomes for all three dice: Total outcomes = 6 * 6 * 6 Total outcomes = 216 The sample space consists of 216 simple events when three dice are tossed.

Key concepts

Sample Space

When discussing probability, the term 'sample space' refers to the set of all possible outcomes or events that can occur during an experiment. For example, when considering three dice being tossed, each die can land on any number from 1 to 6. Therefore, the sample space for one die includes six possible outcomes.

To determine the sample space for three dice, we must consider all possible outcomes for each individual die. Because each die has six sides, we use the multiplication principle to calculate the overall sample space. This principle tells us to multiply the number of possible outcomes for each individual element in the experiment.

Thus, for three dice, the sample space is the product of the outcomes for each die:

• The first die has 6 possible outcomes.
• The second die also has 6 possible outcomes.
• The third die similarly has 6 possible outcomes.

This results in a sample space of 6 * 6 * 6, equaling 216 possible outcomes or simple events.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It's particularly useful in understanding probability by determining the total number of possible outcomes in a given scenario.

In the context of rolling dice, combinatorics helps us calculate the number of ways we can observe different outcomes across multiple dice. Each die tossed is an independent event, with its own unique set of potential results. Using the principles of combinatorics, we multiply the number of possible outcomes for each independent event, as demonstrated with three dice:

• First die options: 6
• Second die options: 6
• Third die options: 6

By applying combinatoric principles, specifically through multiplication, we obtain the total number of combinations possible — which, for three dice, is 216.

Outcomes

An outcome in probability refers to the result of a single experiment or event. When tossing dice, an outcome is the specific number that appears on the top face of a die after it lands. Each die has 6 possible outcomes, corresponding to the numbers 1 through 6.

When multiple dice are used, the total number of outcomes increases exponentially, as shown by multiplying the possibilities for each die together. This means for three dice:

• Outcome for die one could be any of 1, 2, 3, 4, 5, 6.
• Outcome for die two can independently be any of those numbers.
• Similarly, outcome for die three can be any of those numbers as well.

Each combination of these numbers represents a unique outcome, and when considering three dice, the total outcome set includes 216 possibilities. This allows us to evaluate the likelihood of any specific combination occurring during a toss of these dice.