Statistics - Estimation problem I am struggling with a statistics problem that seems quite easy but don't know what to do.
In a factory a product is given to two experts - X and Y. They have to independently test the product and find defects, if any. Expert A found 11 defects and expert Y found 15. After comparison it is known that 8 of the defects are found by both of them. Find an estimation of the number of defects in the product.
The first thing that comes in mind is the mean value - 13. However, I don't know how to use the '8 defects in common' information. Any suggestions?
 A: Without knowing more context, I would say there's 18 defects. Since Expert A found 11 defects but 8 were common with Expert B, Expert A found $11-8=3$ unique defects. Likewise, Expert B found $15-8=7$ unique defects. So we add 3 and 7 to the 8 defects that were found by both, $3+7+8=18$.
A: I'll assume that when a defect exists,
expert $X$ identifies it as a defect with probability $p_X$ and
expert $Y$ identifies it as a defect with probability $p_Y$;
also, neither expert ever identifies something as a "defect"
when it is not actually a defect,
and the experts' chances to identify any actual defect are independent.
Assume there are $N$ defects in the product,
then the most likely outcome is that expert $X$ identifies $p_X N$ defects,
expert $Y$ identifies $p_Y N$ defects, and there are $p_X p_Y N$ defects
that are each identified by both experts.
If we set $p_X N = 11$, $p_Y N = 15$, and $p_X p_Y N = 8$, then
$$p_X = \frac{p_X p_Y N}{p_Y N} = \frac{8}{15},$$
so $$p_X N =\frac{8}{15} N = 11$$
and therefore $N = 165/8 = 20.625.$
You can confirm we get the same result if you estimate $p_Y$ first;
the estimate of $N$ is based on the fact that
$$N = \frac{(p_X N)(p_Y N)}{p_X p_Y N}.$$
The fact that this comes out to a fractional result might seem
unsatisfying, but there is no reason it should be an integer.
What I find actually unsatisfying here is that I am not convinced the
estimate is unbiased.
