The present invention discloses casino baseball gambling games played on electronic video gaming devices including a microprocessor, computer memory chips, programming software or firmware, a video display, control input buttons, and a coin acceptor. In a first embodiment of the invention, multiple players make wagers and compete against each other to win a pot formed by player wagers by scoring the greatest amount of runs in an allotted number of innings of play. Player scores are determined by various simulated "at bat" events such as singles, doubles, strike outs, steals, bunts, and home runs, which occur under program control based upon probabilities for the actual "at bat" events derived from major league baseball statistics. Preferably, players are afforded an option at an intermediate step in game play, after a first one-half inning, to either drop out and forfeit their initial wager, or to continue play by placing a second wager. In a second embodiment, an electronic video slot type machine intended for play by a single player, or by multiple players taking turns in succession, operates under program control to display baseball "at bat" events, associated odds, and separate wagering options on first and second portions of a video display. The second portion of the video display provides an optional "home run" attempt and associated wager which does not affect play of the game on the first display portion.