Suppose I have two biased coins.

I want to estimate the joint probability of both being heads.

SO for coin A, I make n tosses, and have x heads.
for coin B I make m tosses, and have y heads.


For coin A, the best estimate for the probability of a head is

u = x/n

the se for that is

se = sqrt(u*(1-u)/n)

And the 95% confidence bounds are

p = u +/- 1.96 * se.
But how do I get the equivalent bounds for both coins being heads? My
colleague said it was something to do with gaussians, but I can't find
any references on the interweb. I'd like to do similar calculations for more than 3,4,...,n coins.