This activity explores **probability**. The setup:

Biologists often use the mark-and-recapture method as a way to estimate the size of a population. This method involves capturing a sample from the population, tagging each member of the sample, then returning them to the population. A second sample is later captured, and the tagged members are counted and recorded. The results from the two samples can be used to estimate the size of the population.

How does this work, exactly? It’s actually pretty brilliant.

It’s impossible to capture an entire population, whether this means physically trapping animals as in the example or simply knowing the actual number in an experiment like this. The steps to the Mark & Recapture method are essentially thus using a:

- Mark a number of selected individuals in a population, animals or otherwise
- Return those individuals to the natural population
- Allow
*complete*mixing—that is, don’t artificially capture the marked individuals immediately without ensuring they are randomly dispersed back into the population. - Take a
*second*sample. Theoretically, in that second sample, the proportion of marked to unmarked individuals will allow you to determine the entire population using relatively simple math^{1}:

\[ \frac{R \text{ (marked recaptures)}}{T \text{ (total in second sample)}} = \frac{M \text{ (marked initially)}}{N \text{ (total population)}} \]

Knowing this, we can determine the estimated natural population:

\[ N = \frac{M \times T}{R} \]

The actual work for this activity is *analog*. That is, you’ll be using physical objects and recording data on paper. Treat this assignment like *field work* and use a notebook, jotting down your notes and observations as you go.

Create and submit a document based on and including the following:

- Simulate the same procedure using some uniform collection of items such as colored beads, M&Ms (try not to eat any until you’re all done), colored cards, or pieces of paper that you mark with different colors using pens, pencils, markers, etc.
- First, start with a large collection of such items—at least 80 total.
*Just estimate; you wouldn’t actually know the exact number in reality, remember!* - Then, randomly collect a sample of 50 items and use a marker (or something similar) to “tag” each one. Record your results.
- Next, replace the tagged items by mixing them back into the whole population.
- Now, select a second sample to estimate the population size.
- Compare the result to the actual population size by counting all the items. If you used different sample sizes, does your estimate of the population change?
- Test this question by using three different sample sizes.

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