Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. If an individual coin toss or the roll of dice is considered to be a random event, then if repeated many times the sequence of random events will exhibit certain patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.
A handicap (or "odds") in chess is a way to enable a weaker player to have a chance of winning against a stronger one. There are a variety of such handicaps, such as material odds (the stronger player surrenders a certain piece or pieces), extra moves (the weaker player has an agreed number of moves at the beginning of the game), extra time on the chess clock, and special conditions (such as requiring the odds-giver to deliver checkmate with a specified piece or pawn). Various permutations of these, such as "pawn and two moves", are also possible.
Handicaps were quite popular in the 18th and 19th centuries, when chess was often played for money stakes, in order to induce weaker players to play for wagers. Today, except for time odds, handicaps are rarely seen. Rybka, however, a top-rated computer chess engine, played a successful series of handicap matches in 2007 and 2008 against human chess masters.
Fixed-odds betting is a form of wagering against odds offered by a bookmaker, an individual, or on a bet exchange. In Australia the practice is usually known as "SP betting"
The New York Lottery began in 1967 as the third modern U.S. lottery, after Puerto Rico's began in 1934, and New Hampshire's in 1964. It provides revenue for public education, and is based in Schenectady.