Explain the Law of Large Numbers. How does this law apply to gambling casinos? Choose the correct answer below. A. As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to 0 . Casinos use the Law of Large Numbers to determine how many players gamble in certain games. B. As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to 1. This applies to casinos because they are able to make a profit in the long run because they have a small statistical advantage in each game. C. As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. Casinos use the Law of Large Numbers to determine how many players gamble in certain games. D. As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. This applies to casinos because they are able to make a profit in the long run because they have a small statistical advantage in each game.

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**Step 1: Understand the Law of Large Numbers**

The Law of Large Numbers states that as the number of repetitions of a probability experiment increases, the sample proportion of an outcome will approach the true probability of that outcome. In mathematical terms, if $$ p $$ is the true probability of an event, then for many independent trials, the observed frequency $$ \hat{p} $$ will converge to $$ p $$.

**Step 2: Apply the Law to Gambling Casinos**

Casinos design games so that they have a small statistical advantage, which means that the probability of the casino winning is slightly higher than that of the player. Over many repetitions (i.e., many games played by a large number of gamblers), the observed outcomes will converge to the game's expected probabilities. This ensures that in the long run, the casino will make a profit.

**Step 3: Evaluate Each Answer Choice**

- **Choice A:** States that the proportion approaches $$ 0 $$ and relates the law to determining the number of players. This is incorrect because the proportion does not approach $$ 0 $$ but rather the true probability.
  
- **Choice B:** States that the proportion approaches $$ 1 $$. This is also incorrect because the proportion will approach the actual probability, not necessarily $$ 1 $$. While it is true that casinos have a slight advantage, the key point is about the proportion converging to the probability of the outcome.
  
- **Choice C:** States that the proportion approaches the probability of the outcome, but then it incorrectly relates this use to determine how many players gamble in certain games. Casinos rely on many repetitions, not determining the number of players.
  
- **Choice D:** States that the proportion approaches the probability of the outcome and correctly applies this to the fact that casinos make a profit in the long run because they have a small statistical advantage in each game.

**Step 4: Conclusion**

The correct answer is:

**D. As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. This applies to casinos because they are able to make a profit in the long run because they have a small statistical advantage in each game.**
The correct answer is: D. As the number of repetitions increases, the observed outcome approaches the true probability. Casinos use this to ensure they make a profit because they have a slight advantage in each game.

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