Introduction
The opposing football captains watch as the coin arcs, glinting, through the air before landing on the turf at the referee's feet. Heads. Whatchamacallit U. kicks off.
For thousands of years, people have depended on devices that yield outcomes beyond the control of human agency; it has been a way of consulting the gods, or the fates. For us, the point of tossing a coin to determine who kicks off is that the outcome is a matter of chance. The probability of heads is one-half: The coin could, with equal conformity to the laws of the physical world, have landed tails.
Typically, for probability, matters are not as simple as they seem. In ancient times the outcome of chance events—the toss of a knucklebone or a coin—was often taken to reveal the will of the gods. Even today many people take the outcome of a chance event, at least if they have wagered on it, to be a matter of “luck,” where luck plays the role of the old gods, and can be cajoled, sacrificed to, encouraged with crossed fingers and rabbit's feet. In most cases, however, chance events are understood to be outside of human control, and to yield the outcomes they yield in accord with the laws of probability.
The early probabilists (Pascal, Fermat, the Bernoullis, and Laplace) believed in a deterministic world in which chance events did not really exist. Our belief in any chance event (say the outcome of the toss of a six-sided die) is less than certain only as a consequence of the limitations of our knowledge.