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In probability theory, the complement of an event is a fundamental concept that represents all outcomes in a sample space that are not part of the event in question. For example, if we consider rolling a six-sided die, and the event is rolling a number less than 3, the complement of this event would be rolling a number that is not less than 3.

To calculate the probability of the complement of an event, denoted as \(P(E^c)\), we use the formula \(P(E^c) = 1 - P(E)\). This is because the probabilities of an event and its complement must add up to 1, which is the probability of the entire sample space. Therefore, if the probability of rolling a number less than 3 is \(\frac{1}{3}\), then the probability of its complement is \(1 - \frac{1}{3} = \frac{2}{3}\). Understanding this helps students avoid common mistakes when assessing the likelihood of complementary outcomes.

When solving probability problems, it is crucial to consider all the possible outcomes. The probabilities of these outcomes are values between 0 and 1, inclusive, where 0 indicates impossibility and 1 indicates certainty. To express the probability of any outcome, we use the ratio of the number of favorable outcomes to the total number of possible outcomes.

For a six-sided die, there are six possible outcomes (1 through 6). Rolling a number less than 3 includes two favorable outcomes: 1 and 2. Therefore, the probability of rolling a number less than 3 is \(\frac{2}{6} = \frac{1}{3}\). It is key to always simplify fractions to ensure clarity and accuracy in communication of probabilities. By precisely identifying the favorable and total available outcomes, students can more confidently solve probability problems and understand where each outcome fits in the broader context of the sample space.

In the context of validating probability statements, a 'true or false' justification involves a critical examination of the statement based on probability principles and the given data. To justify whether a probability statement is true, one must first understand the event described and then calculate its probability as per the established rules. Following this, the stated probability is compared to the calculated one to affirm its correctness.

In our six-sided die example, the first statement is tested and found true as the probability of rolling a number less than 3 is indeed \(\frac{1}{3}\). However, the second statement is proven false when we identify that the complement of rolling a number less than 3 is actually \(\frac{2}{3}\), not \(\frac{1}{2}\) as stated. Justification in such an exercise improves students' critical thinking skills, and their understanding of probability, and anchors mathematical reasoning in factual calculation rather than assumption.