Analysis of the 50 years of weekly S&P 500 price changes reveals that they are independent of each other with the probability of a positive price change being 0.56. Defining a “success” as a positive price change, let be the number of successes over the next year, that is, over the next 52 weeks. Given the independence of trials, it is reasonable to assume that has the distribution.
Engineers define reliability as the probability that an item will perform its function under specific conditions for a specific period of time. Replacement heart valves made of animal tissue, for example, have probability 0.77 of performing well for 15 years.3 The probability of failure within 15 years is, therefore, 0.23. It is reasonable to assume that valves in different patients fail (or not) independently of each other. The number of patients in a group of 500 who will need another valve replacement within 15 years has the distribution.
Deal 10 cards from a shuffled deck and count the number of red cards. There are 10 observations, and each gives either a red or a black card. A “success” is a red card. But the observations are not independent. If the first card is black, the second is more likely to be red because there are more red cards than black cards left in the deck. The count does not have a binomial distribution.