  The mark-recapture method was devised by Petersen in 1896 to estimate the
number of fish migrating into the Limfjord, and independently by Lincoln in
1930 to estimate waterfowl abundance. The technique applies to any search for a
finite number of items by two or more people or agents, allowing the number of
searched-for items to be estimated. This ubiquitous problem appears in fields
from ecology and epidemiology, through to mathematics, social sciences, and
computing. Here we exactly calculate the moments of the hypergeometric
distribution associated with this long-standing problem, confirming that widely
used estimates conjectured in 1951 are often too small. Our Bayesian approach
highlights how different search strategies will modify the estimates. As an
example, we assess the accuracy of a systematic literature review, an
application we recommend.
