 Solution

The expectation isFor this polynomial to equal $\theta$ for every $\theta \in (0,1)$, comparison of its constant, linear, and quadratic coefficients givesThus the unique choice is

For fly $i$, let $Z_i$ be half its number of $B$ alleles. Then $Z_i\in \{0,1/2,1\}$, the $Z_i$ are independent and identically distributed,and ${\hat{\theta }}_{w^{\ast }}=n^{-1}{\sum }_i{Z}_i$. The weak law of large numbers gives ${\hat{\theta }}_{w^{\ast }}\to \theta$ in probability. Hence the estimator is unbiased and consistent.