Final answer:
A probability distribution for a single fair six-sided die has six outcomes, each with an equal probability of 1/6. The event of rolling at least a five has a probability of 1/3. The long-term relative frequency will approach theoretical probabilities with more rolls, demonstrating the fairness of the dice and concepts like the central limit theorem.
Step-by-step explanation:
To construct a probability distribution for rolling a single die, we consider each outcome of rolling the die, which are the numbers 1 through 6. Since the die is fair, each outcome has an equal probability of occurring. The sample space, S, is {1, 2, 3, 4, 5, 6}. The probability for each outcome is 1/6 since there are six possible outcomes and the die is fair and unbiased. The event E, which is rolling a number that is at least five, has outcomes {5, 6} with a probability P(E) = 2/6 or 1/3.
When a die is rolled multiple times, the long-term relative frequency of these outcomes is expected to approach their theoretical probability. If a die is rolled a few times, outcomes may not align perfectly with their probabilities due to random fluctuations. However, as the number of rolls increases, the observed frequencies are expected to coincide more closely with the expected frequencies if the die is fair.
It's important to note that some dice may be biased due to manufacturing inconsistencies, such as the weight differences created by the holes representing the numbers. In contrast, casino dice are designed to eliminate any such bias and ensure fair play. Through exercises such as rolling multiple dice and calculating sample means, we can observe patterns and verify the fairness of dice, as well as demonstrate the central limit theorem (CLT), which asserts that the distribution of sample means will tend toward a normal distribution as the number of dice (and thus, the number of samples) increases.