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When we talk about probability and experiments like rolling a die, the term **sample space** is crucial. The sample space is essentially the set of all possible outcomes of a given experiment.
For our exercise, this involves rolling a die until a six appears.

A sample space can be simple or complex, depending on the nature of the experiment. In our exercise, the complexity arises because the die is rolled until a desired outcome (rolling a six) occurs.
Hence, the sample space is a collection of sequences where each sequence must end with the number six.

For example:

• (6) – the first roll is a six.
• (1, 6) – the first roll is a one, and the second roll is a six.
• (2, 6) – the second roll is a six after a two on the first roll.

These sequences illustrate different paths through the experiment, as eventually each must conclude with rolling a six. The beauty of the sample space is that it accounts for every potential scenario in this rolling sequence.

To make sense of all these potential outcomes, we often use a **tree diagram**.
A tree diagram visually represents the sample space and helps trace each possible path through an experiment.
For our dice rolling exercise, we start with a single point, the initial roll. This point branches out into six possible outcomes, one for each side of a die.

When a tree diagram is constructed:

• Each branch represents a possible outcome of the die.
• When a "six" comes up, the branch ends since our condition has been met.
• If a roll results in numbers 1 to 5, the experiment continues, and those branches themselves split into another set of six outcomes.

This expansive branching continues until each limb of the tree concludes with a six. The tree diagram not only organizes the outcomes but also illustrates the process of how each outcome is reached through a series of decisions or events.

When engaged in a probability experiment, **experimental outcomes** refer to the final result of a conducted trial. These outcomes are essential as they quantify and qualify the possibilities in our experiment.

In our die-rolling exercise, each sequence of rolls that ends with a six represents a unique experimental outcome. This includes all the possible sequences like (3, 1, 6) or (4, 4, 2, 6), where the outcome finalizes once a six appears.

The idea is straightforward: you keep track of each sequence of numbers rolled until a six is rolled.

• Each outcome tells a story of how many rolls it took to reach that six.
• These outcomes provide insights into the likelihood and patterns observed within the experiment.

Understanding these outcomes is key as it allows one to make predictions or decisions based on probability, lending a structured insight into random events. Each unique path to six embodies a unique set of circumstances, enhancing the richness and depth of our probability study.