[R] Small p from binomial probability function.

[Rolf Turner]

I've figured it out.  It ***is*** "obvious" why Evert's procedure works.
Once you hold your head at the correct angle, as my first year calculus 
lecturer
used to say.

The binom.test() confidence interval gives you the value of a random 
variable
say "U" (for "upper") such that

     Pr(U < p) = p0

where U is a function of the observed binomial random variable, say U = 
h(X).

The observed value of U is h(x), where x is the observed value of X.

Now we want p such that Pr(X <= x) = p0 where X ~ Binom(N,p).

But when X ~ Binom(N,p),

     Pr(U <= p) = p0, i.e
     Pr(h(X) <= p) = p0, so if we take p = h(x) we have
     Pr(h(X) <= h(x)) = p0, whence
     Pr(X <= x) = p0 as desired.

Still twists my head, but.

     cheers,

     Rolf Turner