## Estimating a Population

Estimating the population of a species is desirable, especially when a species is endangered. If is often impractical to track down every member - the species may be migratory, or pr live underground or prefer the darkest holes to hide in. The population is anyway not constant. Breeding is producing young, and the old are dying.
There is a way though to estimate and monitor a changing population. Suppose from a population of a species of total size
$N$
,
$n$
animals are tagged. The fraction of animals that are tagged is
$\frac{n}{N}$
.
The tagged animals are freed, and a second sample of size
$m$
is then taken.
$x$
of these animals are wearing the tag. Assuming the proportion of animals wearing the tag is the same as the proportion of animals that were tagged,
$\frac{n}{N}=\frac{x}{m}$
. Rearranging this gives
$N=\frac{nm}{x}$
.
200 wild pandas are tagged and released, and a year later, a sample of 300 is taken, of which 50 are wearing a tag. The population estimate is then
$N=\frac{200 \times 300}{50}=1200$
.
This estimate assumes that there have not been any births or deaths since the animals were tagged, and the tagged and untagged animals are properly mixed.