Estimate value with binomial distribution We have some compound A diluted in a solution. In 200 trials, we find that when we mix $1$ $\mathrm{mm}^3$ our solution of A with some amount of some compound B, we get a reaction 185 times. How can I find the density of A in the solution?
My first thought was to use an estimator for a binomial distribution, which would have given us the estimate $\hat{p} = 185/200 = 0.925$ but I am lost as to how this can help me find the density of A in the solution.
 A: You do not specify the preconditions for "a reaction". Let us assume that "a reaction" occurs when the drawn sample of $1\mbox{mm}^3$ contains at least one particle of A.
The density $p$ (in terms of particles of A per particles of the dissolver B, not of mass) depends on the number of particles of the dissolver B withim $1\mbox{mm}^3$. Let this number be $N$. Then the probability that your sample of $1\mbox{mm}^3$ does not contain A is $(1-p)^N$. The probability of "a reaction" is the complement, i.e. $1-(1-p)^N$. Your measurement $r=185/200$ is an estimator for this probability. Solving for $p$ yields:
$$r=1-(1-p)^N \quad\Rightarrow\quad \hat{p}=1-(1-r)^{1/N}$$
Edit: When at least m particles of A are required for "a reaction", $(1-p)^N$ must be replaced with the CDF of the binomial distribution, and the resulting equation can only be solved numerically for p, e.g. in R with the builtin function uniroot (r is the measured "reaction rate"):
uniroot(function(p) (1 - r - pbinom(m,N,p)), interval=c(0,1))

