Three sets of rare baseball events--pitching a no-hit game, hitting for the cycle, and turning a triple play--offer excellent examples of events whose occurrence may be modeled as Poisson processes. That is, the time of occurrence of one of these events doesn't affect when we see the next occurrence of such. We modeled occurrences of these three events in Major League Baseball for data from 1901 through 2004 including a refinement for six commonly accepted baseball eras within this time period. Model assessment was primarily done using goodness of fit analyses on inter-arrival data. (Contains 6 tables, 4 figures, and 5 online resources.)