Which of the following is not a valid explanation of the Law of Large Numbers? Choose the correct answer below. A. If one looks at the proportion of times an event has occurred over a long period of time (or over a large number of trials), one can be more certain of the likelihood of its occurrence. B. The relative frequency approximations for an event tend to get better with more observations. C. The more times an experiment is repeated, the closer the relative frequency of an event tends to get to the actual probability of the event. D. These statements are all valid.

The law of large numbers states that the approximation by relative frequency tends to improve as the number of observations increases. This law reflects a fairly simple notion supported by common sense: A probabilistic estimate based on only a few observations may differ greatly, but with a slight difference. Increasing number of observations the estimate tends to be increasingly smaller (precision). With this, we can conclude that it is correct to say that:

If one looks at the proportion of times an event has occurred over a long period of time (or over a large number of trials), one can be more certain of the likelihood of its occurrence.
The relative frequency approximations for an event tend to get better with more observations.
The more times an experiment is repeated, the closer the relative frequency of an event tends to get to the actual probability of the event.

For example, if we do a survey of the population of a Brazilian state and look at only a few citizens (sample), the results may contain a big mistake, but if we analyze several people in several different cities within this state (randomly selected), the results of the samples will be very close to the true values of the population and the larger the sample (larger number of people interviewed) the greater this approximation will be.