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When a decision maker chooses to form his/her own probability distribution by combining the opinions of a number of experts, it is sometimes recommended that he/she should do so in such a way as to preserve any form of expert agreement regarding the independence of the events of interest. In this paper, we argue against this recommendation. We show that for those probability spaces which contain at least five points, a large class of seemingly reasonable combination methods excludes all independence preserving formulas except those which pick a single expert. In the case where at most four alternatives are present, the same conditions admit a richer variety of non-dictatorial methods which we also characterize. In the discussion, we give our reasons for rejecting independence preservation in expert judgement synthesis.