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 *
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 *   To compute the binomial probability, call dbinom(x,n,p).
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 *   To compute the binomial probability, call dbinom(x,n,p).
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 *   This checks for argument validity, and calls dbinom_raw().
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 *   This checks for argument validity, and calls dbinom_raw().
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 *
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 *
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 *   dbinom_raw() does the actual computation; note this is called by
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 *   dbinom_raw() does the actual computation; note this is called by
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 *   other functions in addition to dbinom().
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 *   many other functions in addition to dbinom():
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 *   dnbinom(), dbeta(), df(), dgeom(), dhyper()
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 *     (1) dbinom_raw() has both p and q arguments, when one may be represented
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 *     (1) dbinom_raw() has both p and q arguments, when one may be represented
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 *         more accurately than the other (in particular, in df()).
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 *         more accurately than the other (in particular, in df()).
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 *     (2) dbinom_raw() does NOT check that inputs x and n are integers. This
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 *     (2) dbinom_raw() does NOT check that inputs x and n are integers. This
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 *         should be done in the calling function, where necessary.
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 *         should be done in the calling function, where necessary.
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 *         -- but is not the case at all when called e.g., from df() or dbeta() !
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 *         -- but is not the case at all when called e.g., from df() or dbeta() !