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In this paper we study the amount of secret information that must be given to participants in any secret sharing scheme that is secure against coalitions of dishonest participants in the model of Tompa and Woll [20]. We show that any (k, n) threshold secret sharing algorithm in which any coalition of less than k participants has probability of successful cheating less than some 驴 0 it must give to each participant shares whose sizes are at least the size of the secret plus log 1/驴.