The expected number of events occurring among a collection of events equals the sum of their individual probabilities, proved via indicator random variables and linearity of expectation—a result that holds without requiring independence. This yields Markov-style bounds: the probability that at least one event occurs is bounded above by the expected number of events (useful when that expectation is small), while for mutually independent events with expectation of occurrences large, the probability that none occur is bounded above by e to the negative expected value—a result termed Murphy's Law, since a large expected count of independent events makes at least one occurrence near-certain. The topic also covers the product rule for expectation (requiring independence), the failure of expectation to distribute over ratios or self-products, variance as the expected squared deviation from the mean, and standard deviation as its square root—all within probability theory's treatment of expectation, independence, and dispersion of random variables.
The expected number of events occurring among a collection of events equals the sum of their individual probabilities, proved via indicator random variables and linearity of expectation—a result that…