The Law of Large Numbers
John E. Freund ( Introduction to Probability ) has been discussing topics like the "odds" that an airline flight from Chicago to Los Angeles will arrive on time. He says that if 688 of the last 800 flights have been on time, we can say the probability of this flight being on time is .86. Then he asks, "When probabilities are thus estimated, it is only reasonable to ask whether the estimates are any good. The answer, which is 'Yes,' is supported by a remarkable law called the Law of Large Numbers ... Informally, this law can be stated as follows: " If the number of times the situation is repeated becomes larger and larger, the proportion of successes this will tend to come closer and closer to the actual probability of success. " Later on, he states this law formally: " If a random variable has the binomial distribution, the probability is at least 1 - 1 / k 2 that the proportion of successes in n trials will differ from p by less than k *...