The method of choosing and distributing enhanced odds includes 1) linking numerous electronic gaming devices, such as electronic Poker machines, on a local network, 2) gathering in a progressive amount a percentage of all the bets placed on those devices such as for progressive jackpots, 3) using a process based on a combination of randomness and criteria to distribute for at least one game to one randomly selected device, a set of enhanced odds and 4) using the progressive amount gathered to pay prizes awarded with the set of enhanced odds. In a local network of electronic Poker machines (2), five percent (5%) of the bets are gathered by the network. When a predetermined amount is gathered, one electronic Poker machine randomly selected, presents to a player one game with a set of enhanced odds. When a win occurs corresponding to one of the four (4) highest hands, the prize is paid in regards to the enhanced odds.