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The concept of a sample space is a fundamental principle in probability theory. It refers to the complete set of all possible outcomes of a random experiment. In the context of tossing a fair coin three times, each toss can yield one of two results: heads (H) or tails (T).

To determine the sample space, we consider all possible sequences of these results over the three coin tosses. Thus, the sample space comprises every combination of H and T, such as HHH, HHT, and so forth. Specifically, for three coin tosses:

• Each toss can result in H or T.
• Since there are three tosses, we evaluate all possible sequences of these results.
• This results in a total of 8 possible outcomes, as calculated by the formula \(2^3\). This is because there are two possible outcomes (H or T) for each of the three tosses.

Consequently, the sample space \( S \) for this process is: \[ S = \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \} \].

Each sequence in this set represents a unique potential outcome when tossing the coin three times.

A fair coin is a primary assumption in probability exercises. It implies that the coin has no bias, meaning each side—heads or tails—has an equal probability of landing face up. This quality of fairness is crucial for determining probabilities accurately in an experiment.

When we talk about a fair coin, we mean:

• Each side of the coin is equally weighted.
• The chance of obtaining heads is the same as tails, which is \(\frac{1}{2}\).

The fairness of the coin ensures that all outcomes in our sample space are equally likely to occur. In our case of three tosses:

• Each of the 8 sequences (like HHH, HTT, etc.) has the same likelihood of occurring because of the coin's fairness.
• This equality simplifies the calculation of probabilities for each outcome, as each sequence is equally probable.

Examining this concept helps students understand how assumptions about fairness can affect the outcomes and probability calculations in probability theory.

Once we've established our sample space, the next step is to assign probabilities to each outcome. In probability theory, this assignment reflects how likely each outcome is compared to the others.

Given that we are using a fair coin, each possible outcome when tossing the coin three times occurs with equal likelihood. Understanding this, we can assign a probability of \(\frac{1}{8}\) to each sequence because:

• There are 8 possible outcomes in our sample space.
• Each outcome is the result of three successive tosses, with each toss having a probability of \(\frac{1}{2}\).
• Thus, the likelihood of any specific sequence occurring—like HHH or TTH—can be calculated as \((\frac{1}{2}) \times (\frac{1}{2}) \times (\frac{1}{2}) = \frac{1}{8}\).

This uniform probability assignment hinges on the fair nature of the coin, which guarantees no bias in the results. The process shown highlights how, in experiments involving equally likely outcomes, probability is distributed uniformly across all potential sequences.