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The concept of a sample space is crucial in probability theory because it includes all possible outcomes of an experiment. In a coin toss, each flip can result in either heads (H) or tails (T). When you toss a coin three times, the sample space consists of all sequences of heads and tails that could result. For instance, if we denote heads by H and tails by T, the three-toss sample space is: \(S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}\). Understanding the sample space helps us calculate probabilities for different events.

Event probability refers to the likelihood of a specific event happening based on the sample space. For example, if we want to find the probability of getting tails on the first toss, we determine how many outcomes in the sample space match this event. In our three-toss scenario, the outcomes with tails on the first toss are \(\{THH, THT, TTH, TTT\}\). Thus, the probability of getting tails on the first toss (\(E_1\)) is calculated by dividing the number of favorable outcomes by the total number of outcomes: \(p(E_1) = \frac{4}{8} = \frac{1}{2}\). By similarly finding the probability of other events, we can analyze more complex sequences, such as getting heads on the second toss or getting two heads but not in a consecutive row.

The intersection probability, denoted as \(p(E_1 \cap E_2)\), measures the probability that two events both happen. To calculate this, we identify the outcomes that are common to both events. For example, let's consider two events: \(E_1\) is getting tails on the first coin toss, and \(E_2\) is getting heads on the second toss. The outcomes where both \(E_1\) and \(E_2\) occur simultaneously are \(\{THH, THT\}\). Thus, \(p(E_1 \cap E_2)\) is the probability of these combined outcomes: \(p(E_1 \cap E_2) = \frac{2}{8} = \frac{1}{4}\). Intersection probability helps us understand the relationship between different events, especially when determining whether they are independent.

Analyzing outcomes of a coin toss helps in understanding fundamental probability concepts. When a coin is tossed multiple times, each sequence represents a unique outcome. For instance, tossing a coin three times yields \(2^3\) or 8 different sequences: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Each sequence has an equal chance of occurring if the coin is fair, making the probability of any specific outcome \(\frac{1}{8}\). Events in this context can be getting tails on one of the tosses, or having heads appear in a particular position. Breaking down these outcomes and their respective probabilities lays the foundation for comprehending more complex probability concepts like sample space, event probability, and intersection probability.