Background
I've recently read the paper
Leo A. Goodman, On the Exact Variance of Products
Journal of the American Statistical Association
Vol. 55, No. 292 (Dec., 1960), pp. 708-713
from where I extract the following edited quotes (removed superfluous calculations and sentences)
Let $x$ and $y$ be two independent random variables. Let us denote the expected value of x by $E(x) = X$, the variance of $x$ by $V(x)$, ... A similar notation will be used for the random variable $y$.
...we have that the variance $V(xy)$ of the product $xy$ is equal to $$ V(xy) = \ldots = X^2V(y) + Y^2V(x) + V(x)V(y)$$
... We shall now present an unbiased estimate of the variance $V(xy)$. ... we have that $$ v(xy) = \ldots = x^2v(y) + y^2(x) - v(x)v(y)$$
is an unbiased estimate of $V(xy)$, where $v(x)$ is an unbiased estimate of $V(x)$ and $v(y)$ is an unbiased estimate of $V(y)$.
I have a relatively simple formula $P = w + xy/(1-z)$ where each of these (independent!) variables have been estimated by a statistical package, and supplied along with 95% confidence limits and standard errors (hence variances). In fact, each of $w,x,y,z$ are probabilities, and $z$ is bounded away from 1. (as an example of the magnitudes involved, one instance of the problem has $0.1 \lt w,x,y,z \lt 0.6$ and all standard errors about $3 \times 10^{-3}$)
Questions
I need to estimate some confidence limits on $P$, and my first idea was to use the confidence limits of $w,x,y,z$, but it looks tricky/inadvisable. My second idea was to work out the variance of $P$. This clearly boils down to finding the variance for $xy/(1-z)$.
Someone has told me that I should use the equation for $v(xy)$ in the context of my formula. That is all well and good, I can accept that. So now all I need to do is find the variance of $1/(1-z)$ and apply the result of the Goodman paper twice, or perhaps only find the variance of $y/(1-z)$ and use the Goodman result once. For argument's sake, let's do the former.
I found on the internet a rough set of notes which estimated the variance of a ratio $x/y$ to be (taking the special case of $x,y$ independent) $$ Var(x/y) \approx \frac{E(y)^2 Var(x) + E(x)^2 Var(y)}{E(y)^4} $$ and for the case that I am interested in, I can take $x \sim Uniform(0,1)$ (i.e. '$1$') and so get $$ Var(1/y) \approx \frac{Var(y)}{E(y)^4} \quad \quad (1) $$ Is this reliable/right? Even if it is, I now am faced with a small conundrum. What is the analogue in this instance for the formula for $v$?
I am happy to take all answers that address my original problem, the question of approximating $Var(1/(1-z))$, whether I use $Var$ as given in the approximation (1) or some "unbiased estimate" in terms of the data I do have, and lastly, what would this "unbiased estimate" be, given (1)?