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xCareful analysis leads to the conclusion that security is unrelated to whether the software is proprietary or open source.
xThe open-source movement is largely devoid of systematic efforts to guarantee security.  The fact that code can be examined for flaws does not mean it will be examined by anyone competent.
xThe literature contains reports of serious security flaws in open-source products, often after years of use.  Several occurred in the parts of the software intended to make it secure, which presumably underwent more careful coding and examination.  That strongly suggests that either the many people who supposedly look at the code are not able to recognize the problems, or they aren't really looking.  Experience indicates that both are true.