Biologists doing studies in a particular environment often...

Biologists doing studies in a particular environment often tag and release subjects in order to estimate the size of a population or the prevalence of certain features in the population. Ten animals of a certain population thought to be extinct (or near extinction) are caught, tagged, and released in a certain region. After a period of time, a random sample of 15 of this type of animal is selected in the region. What is the probability that 5 of those selected are tagged if there are 25 animals of this type in the region?

Key Concepts

Hypergeometric Distribution

This concept is used to calculate the probability of obtaining a fixed number of successes in a sample drawn without replacement from a finite population that contains a known number of successes. It applies when the probability of success changes on each draw because the items are not replaced, as is the case in most tagging experiments.

Sampling Without Replacement

This concept involves selecting individuals from a finite set where each individual is not returned to the population after being selected. This alters the probabilities for subsequent selections and is crucial for accurately determining the likelihood of certain outcomes in scenarios like tagging and recapturing specimens.

Combinatorics

Combinatorics is the branch of mathematics dealing with the counting, arrangement, and combination of objects. In this context, it helps in calculating the number of ways to choose a certain number of tagged and untagged animals from the population, which is essential for applying the hypergeometric distribution formula.

Capture-Recapture Method

This is a common ecological technique used to estimate the size of animal populations. It involves capturing a number of individuals, marking them, releasing them back into the environment, and then capturing a second sample. The proportion of marked individuals in the second sample provides an estimate of the overall population size.

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