In order to estimate the population of fish in a lake, a biologist catches 75 fish, marks them, releases them. Next month she catches 75 fish and finds that 5 are tagged. Use this to estimate the fish population in the lake.

Estimating the total number of fish in a lake without draining it might seem like an impossible task, but biologists have a clever method called mark-recapture, also known as the Lincoln-Petersen index. This technique relies on the principle of proportions. Imagine the initial 75 fish caught and marked represent a known fraction of the entire lake's population. When a second sample of fish is caught later, the proportion of marked fish in that sample should ideally reflect the proportion of marked fish in the entire lake.

In this scenario, 75 fish were initially marked and released. In the subsequent month, another 75 fish were caught, and 5 of them had tags. This means that 5 out of 75 fish in the second sample were marked, which simplifies to 1/15. If 1/15 of the second sample is marked, we can assume that roughly 1/15 of the entire fish population in the lake is also made up of those initial 75 marked fish. To find the total population, we can set up a simple proportion: (number of marked fish in first catch / total population) = (number of marked fish in second catch / total fish in second catch). Plugging in the numbers, 75 / N = 5 / 75. Solving for N (the estimated total population), we get N = (75 * 75) / 5, which equals 1125.

The mark-recapture method is a cornerstone in wildlife biology for estimating population sizes of mobile animals that are difficult to count directly. It's widely used for various species, from insects to large mammals, and is crucial for conservation efforts and understanding population dynamics. However, for the estimate to be accurate, certain assumptions must hold true, such as the marked fish mixing randomly with the unmarked population, the marks not affecting the fish's survival or behavior, and no significant births, deaths, immigration, or emigration occurring between the two sampling periods. Researchers carefully design their studies, considering factors like marking methods and the time between captures, to ensure the reliability of their population estimates.