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A fair six-sided die is rolled \(n\) times. Let \(X_i\) be the outcome of the \(i\)-th roll, and let \(\overline{X}_n\) be the average value of the rolls after \(n\) trials.
Calculate the theoretical mean \(\mu\) of a single roll.
According to the Law of Large Numbers, what value does \(\overline{X}_n\) approach as \(n\) becomes very large?
If you roll the die 10 times and get an average of 4.2, does this disprove the LLN? Explain.
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