Gaming system and method having configurable bonus game triggering outcomes

A gaming system and method which provides for one or more gaming device triggered bonus games to be skipped or otherwise not provided to the players at the gaming devices which triggered such bonus games. The average expected payouts of skipped gaming device triggered bonus games are contributed to a bonus accumulation pool or fund. The accumulation pool is subsequently utilized to provide one or more gaming system triggered bonuses to one or more players at one or more of the gaming devices in the gaming system. A player's accumulated triggering symbols determine, at least in part, the player's probability of winning part of the bonus accumulation pool and/or the player's portion of the bonus accumulation pool.