102 Chapter 4 Historically, the first rigorous application of probability theory involved games of chance. Today, casinos use probabilities when they set payoffs for roulette, craps, and slot machines. Governments use probabilities when they set payoffs for state lotteries. Mathematicians use probabilities to devise optimal strategies for blackjack, backgammon, Monopoly, poker, and many other games. Devoted players can learn these probabilities firsthand from long and sometimes expensive experience. Another early use of probabilities was in setting insurance rates. This is why life insurance premiums depend on whether a person is 18 or 98, is in good health or using an artificial heart, or is a college professor or a soldier of fortune. Probabilities are used to price medical insurance, car insurance, home insurance, business insurance, and even insurance against a baseball strike or a singer getting laryngitis. Colleges use probabilities when they decide how many students to admit, how many professors to employ, how many classrooms to build, and how to invest their endowment. Probabilities are involved when an army decides to attack or retreat, when a business decides to expand or contract, and when you decide whether or not to wear a raincoat. Uncertainties are all around us and so are probabilities. 4.1 Describing Uncertainty