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Imagine tossing a coin multiple times. The sample space is essentially a list of all possible outcomes that might result. For a single toss of a coin, we have two potential outcomes: heads (H) and tails (T). Hence, the sample space for one coin toss is \( S_1 = \{H, T\} \).

When we extend this to three tosses of a coin, we combine each result independently. This means that for each of the three tosses, the outcome can either be H or T. So, the sample space for three tosses becomes a sequence of outcomes. In total, we have \( 2^3 = 8 \) different possible combinations or sequences because each of the three tosses has two possibilities. Therefore, the sample space for three tosses is \( \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \).

Here are the sequences you might encounter about sample spaces:

• "HHH" corresponds to all heads.
• "HTH" implies a head, then a tail, then a head.
• "TTT" means all tails.

Each outcome listed represents a possible scenario from the experiment.

Independent events are events that do not influence each other. This is significant when considering multiple tosses of a coin. If we are tossing a fair coin more than once, each toss is independent of the previous ones. The result of one toss does not affect the outcome of the next toss.

For example, if you toss a coin and it lands on heads, the probability of heads on the next toss is still the same: 0.5. This independence allows us to calculate the probability of outcomes across multiple events by multiplying the probabilities of each independent event.

In our exercise, there are three tosses of the coin. Each toss is independent from the others, meaning each outcome, H or T, remains equally probable regardless of previous results. Hence, the probability of getting any particular sequence is computed by multiplying the probabilities for each toss: \( P(\text{sequence}) = 0.5 \times 0.5 \times 0.5 = 0.125 \).
Understanding this concept helps in scenarios where independent events occur, such as rolling a die, picking a card from a shuffled deck multiple times without replacement, etc.

A fair coin is a fundamental concept in probability. It is a simple, unbiased mechanism where there are only two possible outcomes: heads (H) or tails (T). Each outcome, whether heads or tails, has an equal likelihood of occurring. That makes the probability of either landing at 0.5, or \( P(H) = 0.5 \) and \( P(T) = 0.5 \).

This fairness is an assumption used in exercises and experiments involving coin tosses, ensuring that the outcomes are completely random and not skewed in favor of one result. This assumption is crucial when calculating probabilities over multiple events because it guarantees equal chances, allowing precise mathematical modeling of the situation.

Knowing how a fair coin operates is foundational for understanding more complex probability scenarios, as it provides a simple example of symmetry and balance between outcomes.

Outcome probability refers to the likelihood of a specific result from a chance process. When dealing with a fair coin, each toss is a random event with two possible outcomes. If we toss the coin three times, we need to calculate the probability that any specific outcome occurs.

Since each outcome from our sample space in a three-toss scenario has the same probability due to the fairness of the coin and independence of each toss, we use multiplication. The probability for any one sequence is determined as follows: \( P(\text{outcome}) = 0.5 \times 0.5 \times 0.5 = 0.125 \).

This means that each of the sequences in the sample space (e.g., HHH, HHT, HTH, etc.) has an equal probability of occurring: 0.125. This uniform assignment of probability is typical for independent, fair events, as it distributes the likelihood equally across all potential outcomes. Understanding outcome probability aids in tackling more complex probability models, such as those involving dice, spinners, or card games.