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In probability theory, the sample space is the set of all possible outcomes of a random experiment. When dealing with a discrete scenario, such as tossing coins, we need to consider all the potential results.

For the exercise where a penny and a nickel are tossed, each coin can either land as heads (H) or tails (T). Therefore, the combination of outcomes from both coins forms the sample space. This is represented as an ordered pair where the first entry denotes the penny and the second the nickel.

Given this setup, the sample space, denoted by \( \Omega \), includes the following outcomes:

• \((H, H)\): Heads on both the penny and nickel.
• \((H, T)\): Heads on the penny, tails on the nickel.
• \((T, H)\): Tails on the penny, heads on the nickel.
• \((T, T)\): Tails on both the penny and nickel.

Understanding the sample space is crucial as it provides the foundation upon which we determine the probabilities of different events.

Independent events in probability are those whose occurrence does not influence each other. In simpler terms, the outcome of one event does not affect the probability of another. When examining independent events, their combined occurrence probability is the product of their individual probabilities.

Using the events defined in the exercise:

• Event \( A \): Heads on the penny
• Event \( B \): Tails on the nickel
• Event \( C \): Coins landing the same way

To check if any pair of these events is independent, verify if the probability of their intersection equals the product of their probabilities. However, as shown in the solution, none of the event pairs \((A, B), (A, C), \text{and} (B, C)\) satisfy this condition. Thus, these events are not independent. Understanding independent events helps in simplifying complex problems where multiple events occur together.

The intersection of events involves finding the outcomes common to multiple events. In probability, the intersection of two events, say \(A\) and \(B\), is denoted by \(A \cap B\) and refers to outcomes that satisfy both events occurring simultaneously.

In this exercise:

• For events \(A\) and \(B\), the intersection \(A \cap B\) includes outcomes where both heads show up on the penny and tails on the nickel, resulting in \(\{(H, T)\}\).
• Between \(A\) and \(C\), intersecting means both heads must be on the penny, with both coins showing heads, leading to \(\{(H, H)\}\).
• Lastly, for \(B\) and \(C\), tails show up on the nickel with both coins showing tails, forming \(\{(T, T)\}\).

The intersection of events is a valuable concept as it allows us to compute probabilities of simultaneous occurrences in a structured manner. It also informs whether these events are independent by comparing their intersections to the expected probability products if they were independent.