Probability only comes into play directly when you are dealing with models that have imperfect information (such as typical card games where cards are hidden,) some form of stochasticity (any type of random number generation such as shuffling or dice rolling,) or incomplete information (such as Dilemmas, where the competing agent's decision-making process is unknown, although incomplete information is something of a blanket term often applied to any form of inaccessible information.)
Game theory, via minimax, can be applied to models without these conditions, such as non-chance, perfect information games, without requiring probabilities. The major caveat here is that if the model is intractable, and the search space (gametree or, ideally a graph) cannot be exhausted, probability analysis becomes critical--all of the cutting-edge Artificial Intelligence methods rely heavily on statistics.
Logic is important, but it's probably a good idea to at least have some grounding in probability and statistics. The above links should provide pointers to general game theory references.
For probability, I might recommend starting with Bayes. Russell and Norvig's Artificial Intelligence: A Modern Approach is highly regarded [See: Part IV Uncertain Knowledge and Reasoning]. (Here is a link to a pdf of the second edition if you want to check it out, although the 3rd edition is the most current.)