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When discussing probability, 'equally likely outcomes' refers to outcomes that have the same chance of occurring. This concept is fundamental when dealing with experiments like rolling a die or tossing a coin.
For instance, when you roll a fair six-sided die, each of the six faces (1, 2, 3, 4, 5, and 6) has an equal probability of appearing, which is \( \frac{1}{6} \). Similarly, when you toss a fair coin, it has two equally likely outcomes: heads (H) and tails (T), each with a probability of \( \frac{1}{2} \).
Understanding that these outcomes are equally likely is crucial in constructing and analyzing sample spaces.

Rolling a die is a common experiment in probability. A standard die has six faces, numbered 1 to 6.
When you roll the die, you have six possible outcomes. Each number from 1 to 6 is an outcome. Because the die is fair, each of these outcomes is equally likely.
To represent this, you simply list all possible outcomes: \[ \{ 1, 2, 3, 4, 5, 6 \} \].
Here, every outcome has a probability of \( \frac{1}{6} \). This is an essential part of creating a well-defined sample space, since it lays the foundation for any combined experiments involving a die.

A coin toss is another fundamental experiment in probability. It involves flipping a coin to see which face lands upward. A fair coin has two faces: heads (H) and tails (T).
Each face is equally likely to appear when the coin is tossed. This means both H and T have a probability of \( \frac{1}{2} \).
To list the possible outcomes of a coin toss, you write down: \[ \{ H, T \} \].
These outcomes form a simple sample space, but are foundational when combining with other experiments.

When combining two simple experiments, such as rolling a die and tossing a coin, you need to consider all possible outcomes together. This is called a combined experiment.
First, identify the sample spaces for each individual experiment: rolling a die \[ \{ 1, 2, 3, 4, 5, 6 \} \] and tossing a coin \[ \{ H, T \} \].
To find the sample space of the combined experiment, pair each outcome of the die with each outcome of the coin. This results in 12 equally likely outcomes: \[ \{ (1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T) \} \].
Each pair represents a unique combination of rolling the die and then tossing the coin, ensuring all possible scenarios are covered.