Let $X_1, ..., X_n$ i.i.d. from $X$ where $X$ is binary and we have p = P{$X=1$} ,$p\in (0,1)$. Let $\alpha = \frac{p}{1-p}$. Define $\bar{a} = \frac{\bar{X}}{1-\bar{X}}$, show that $\bar{\alpha}$ is unbiased and consistent for $\alpha$.
So to prove unbiased I show: $E[\bar{X}] = E[X]$ since we have $X$ binary, $p = E[X]$. By the continuous mapping theorem since the numerator $=p$ in probability and the denominator $=1-p$ we have unbiased $\bar{\alpha}$.
To prove consistency, we note $\bar{X} \rightarrow E[X]$ (in probability) by weak law of large numbers and the same for $1-\bar{X}$. By continuous mapping, $\alpha$ is consistent.
I feel like there there are flaws in my arguments somewhere.