Question

1. (4 points) how do you think the relative frequency probabilities would compare to the theoretical probabilities if we rolled the pair of dice 500 times? 1,000 times? 1,000,000 times?

Explanation:

Apply the Law of Large Numbers

$$ \lim_{n \to \infty} P_{\text{empirical}}(n) = P_{\text{theoretical}} $$

Compare sample sizes

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Answer:

As the number of rolls increases from 500 to 1,000, and then to 1,000,000, the relative frequency probabilities will become increasingly closer to the theoretical probabilities. This is governed by the Law of Large Numbers, which states that as the number of trials of a random experiment increases, the empirical probability (relative frequency) approaches the theoretical probability.