AbstractSome£400m was estimated as having been staked on gaming machines ("fruit" machines or one-armed bandits) in Great Britain during 1976, making them the form of gambling with the third highest turnover after casinos and off-course betting. Yet little work has been done on their properties, particularly of modern sophisticated machines which incorporate so-called "features" such as "HOLD", "GAMBLE" and "NUDGE" designed to make them more interesting as a gambling medium. The difficulties involved in assessing the properties of such machines, particularly those resulting from the inclusion of a "HOLD" mechanism, are discussed. The basic random variables needed to describe their behaviour are defined; the consequent problems of determining the "expected pay-out" of a machine, both for the purposes of describing its properties to players and for testing the performance of the machine, are investigated. This work is placed in the context of the recommendations of the recent report of the Royal Commission on Gambling.