Since a baseball game consists of a sequence of half-innings commencing in an identical manner, one is tempted to suppose that the progress of the score throughout a game would be well simulated by a sequence of random drawings from a single distribution of half-inning scores. On the other hand, the instincts of a baseball fan are offended at so simple a suggestion. The hypothesis is examined by detailed analysis of 782 professional games, and a supplementary analysis of a further 1000 games. It is shown that the scoring does vary significantly in the early innings, so that the same distribution cannot be used for each inning. But, with a few exceptions, total scores, establishments of leads, overcoming of leads, and duration of extra-inning games as observed in the actual games show good agreement with theoretical calculations based on random sampling of the half-inning scoring distributions observed.