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?
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.