Question

which situation demonstrates the law of large numbers?
if you roll a six - sided number cube 36 times, you will roll a 4 six times.
if 800 people are randomly surveyed about the tv shows they watch,
about 400 of them will be women.
if you flip a coin 44 times, you will get heads about 22 times.
if it snowed four days last month, it will snow four days this month.

Explanation:

Step1: Recall the law of large numbers

The law of large numbers states that as the number of trials increases, the experimental probability approaches the theoretical probability.

Step2: Analyze each option

• For the first option: Rolling a six - sided cube 36 times. The theoretical probability of rolling a 4 is \(\frac{1}{6}\). \(36\times\frac{1}{6} = 6\). But the law of large numbers is about the approximation as the number of trials gets large (not just 36, which is a relatively small number in the context of the law of large numbers).
• For the second option: Surveying 800 people. If we assume equal probability of men and women (theoretical probability of a woman being \(\frac{1}{2}\)), \(800\times\frac{1}{2}=400\). A sample size of 800 is relatively large, and the result is based on the theoretical probability (\(\frac{1}{2}\)) multiplied by the number of trials (sample size).
• For the third option: Flipping a coin 44 times. The theoretical probability of getting heads is \(\frac{1}{2}\). \(44\times\frac{1}{2}=22\). But 44 is a relatively small number for the law of large numbers.
• For the fourth option: Snowing 4 days last month and predicting 4 days this month. There is no probability - based theoretical foundation (like a fair coin or a fair die) for this prediction.

Answer:

If 800 people are randomly surveyed about the TV shows they watch, about 400 of them will be women.