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Registered voters are individuals who have completed the necessary legal requirements to participate in elections by casting their vote. In our exercise, there are 972 registered voters out of 1265 eligible voters. This means that these 972 individuals have taken steps to ensure that they can have a say in the electoral processes of their town.

• Eligibility vs. Registration: Being eligible means that a person meets the basic citizenship, residency, and age criteria. However, registration is the formal step one must take to actually vote.
• Importance: Registration is crucial as it usually ends before election day, ensuring a smooth voting process. Without registration, eligible voters cannot exercise their right.

By dividing the number of registered voters by the total number of eligible voters, we can find the probability that a randomly chosen eligible voter is registered. This calculation is fundamental in determining how representative a sample of registered voters is within a larger group of eligible voters.

In probability, a sample space is the complete set of all possible outcomes that could occur in a particular experiment. For our exercise with voters, the sample space includes two events: a voter either being registered or not registered. Simple as it sounds, specifying the sample space is a key step because:

• It ensures that we've accounted for all possible scenarios.
• It helps in calculating probabilities accurately, as probabilities are simply measures of the likelihood of events within the sample space.

For the problem, this means our sample space consists of:

• Being Registered: This is one outcome from the sample space of selecting an eligible voter.
• Not Being Registered: This is the complementary event to being registered.

Listing these outcomes ensures we understand the scenario fully and are ready to perform the necessary probability calculations.

Complementary events are a pair of events where the occurrence of one means the non-occurrence of the other. In the context of voter registration in our exercise, two events form a complementary pair:

• Event A: The event a voter is registered.
• Event B: The event a voter is not registered.

These two events are mutually exclusive. This means they cannot occur at the same time. If a voter is registered, they cannot simultaneously be not registered, and vice versa.

Because these events are complementary, their probabilities add up to 1. Mathematically:\[ P(Registered) + P(Not \, Registered) = 1 \]This relationship is fundamental because it reflects that our sample space (registered and not registered) covers all possible outcomes. Understanding complementary events aids in comprehending event relationships and simplifying probability problems.

Probability calculations allow us to determine the likelihood of certain events occurring. In the exercise, calculating the probability of an eligible voter being registered involves simple division. We compute it as follows:

• Probability Formula: \[ P(Registered) = \frac{Number \ of \ Registered \ Voters}{Total \ Number \ of \ Eligible \ Voters} \]
• Substitute the values: \[ P(Registered) = \frac{972}{1265} \approx 0.768 \]

This tells us that there is approximately a 76.8% chance a randomly selected eligible voter will be registered.

For the complementary event (voter not being registered), we calculate:

• Complementary Formula: \[ P(Not \, Registered) = 1 - P(Registered) \]
• Calculate: \[ P(Not \, Registered) = 1 - 0.768 = 0.232 \]

Thus, there is about a 23.2% probability that the voter is not registered.

This straightforward method of probability calculation helps us evaluate real-life scenarios effectively, providing insight into how samples behave in larger populations.