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A die is a cube with dots on each face. The faces have \(1,2,3,4,5,\) or 6 dots. The table below is a computer simulation (from the software package Minitab) of the results of rolling a fair die 20 times. $$\begin{array}{c|cccccccccc} {\text { DATA DISPLAY }} \\ \text { ROW } & \text { C1 } & \text { C2 } & \text { C3 } & \text { C4 } & \text { C5 } & \text { C6 } & \text { C7 } & \text { C8 } & \text { C9 } & \text { C10 } \\ \hline 1 & 5 & 2 & 2 & 2 & 5 & 3 & 2 & 3 & 1 & 4 \\ 2 & 3 & 2 & 4 & 5 & 4 & 5 & 3 & 5 & 3 & 4 \end{array}$$ (a) Assume that each number in the table corresponds to the number of dots on the upward face of the die. Is it appropriate that the same number appears more than once? Why? What is the outcome of the fourth roll? (b) If we simulate more rolls of the die, do you expect to get the same sequence of outcomes? Why or why not?

Step by step solution

01

Understand Dice Roll Possibilities

A fair die has six faces, each displaying one of the numbers 1 through 6. Therefore, when you roll a die, the outcome can be any of these six numbers.

02

Check for Multiple Occurrences

In probability, when you roll a fair die multiple times, each roll is independent of others. It is perfectly normal for the same number to occur multiple times in a series of rolls.

03

Identify the Fourth Roll Outcome

For question (a), determine the outcome of the fourth roll from the given table. According to the table, the fourth roll is represented by 'C4' in the first row, which has a value of 2.

04

Consider Independent Events

For question (b), each roll of a fair die is an independent event. Even with multiple simulations, you should not expect the exact same sequence of outcomes due to the random nature of rolling a die.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events

When discussing probabilities, understanding the concept of **independent events** is crucial. Consider rolling a fair die. Each face has an equal chance of landing up, ranging from 1 through 6. What's essential to remember, is that each roll of the die is independent of the others.
This means that the outcome of one roll does not influence the outcome of another. For example, if you roll a die and get a 4, the chance of rolling a 4 on the next roll stays the same. Independent events are characterized by having no effect on one another:

• Each roll is a separate event, unconnected to what came before.
• The probability of rolling any given number remains unchanged, no matter the outcomes of previous rolls.

This independence forms the foundation of understanding complex probabilities, and helps clarify why similar numbers can appear multiple times in a sequence. Since each roll is independent, a sequence can contain the same number repeatedly, without indicating any pattern or predictability.

Simulation

**Simulation** is a valuable technique used in probability and statistics to mimic real-world processes. It is especially useful when you want to analyze scenarios with multiple outcomes without performing the actual experiment repeatedly. In the exercise, we see a simulated roll of a die using software like Minitab. Simulation tools imitate the action of rolling a die, generating results in an instant:

• They save time compared to physical experiments.
• They allow for large datasets, making it easy to observe frequency and patterns in outcomes.

Although a simulation provides a practical approach to understanding probabilities and outcomes, keep in mind it still operates under the same rules of randomness and independence that govern actual dice rolls. This means every simulation results in a different sequence of numbers, just as real dices would create different sequences upon every new series of rolls.

Randomness

**Randomness** plays a crucial role in probability and the outcomes of events like dice rolls. The essence of randomness is the unpredictability of each occurrence, meaning you can never know for sure what the outcome will be before it happens. When you roll a fair die, each number between 1 and 6 has an equal chance of landing up. This randomness ensures that outcomes are unpredictable across multiple rolls:

• It is the reason you can't expect the same outcome sequence in repeated tests.
• Even with simulation, every run will create a different order of numbers.

Randomness ensures fairness in games and simulations, making it central to activities involving chance. It highlights that while methods and tools can predict the frequency of outcomes over many trials, they cannot predict specific sequences or results due to the inherent chaotic nature of random events.