Let N be the total population. The proportion of tagged bees in the second sample should approximate the proportion in the whole population: 200 / N = 36 / 180.
["Using Proportional Sampling to Estimate Total Bee Population with Tagged Individuals", "Estimating the total population of wild bees in a habitat is essential for ecological monitoring, conservation efforts, and understanding ecosystem health. A powerful and simple method for this estimation is proportional sampling—relying on the principle that the proportion of tagged individuals in a sample accurately reflects their proportion in the entire population.", "### The Fundamental Proportionality", "In sample-based population estimation, the proportion of tagged bees in a second sample should closely approximate the proportion of tagged bees in the total population. Mathematically, this is expressed as:", "[\n\frac{\ ext{Number of tagged bees in sample}}{\ ext{Sample size}} \approx \frac{\ ext{Total tagged bees}}{\ ext{Total population}}\n]", "This concept is the foundation of mark-recapture techniques widely used in ecology.", "### The Given Ratio: 200 Tagged Bees in Sample, 36 in Second Sample of 180 Bees", "Let:\n- ( N = \ ext{total bee population (unknown)} )\n- 200 bees were tagged and released\n- In a second sample of 180 tagged bees, 36 are found to be tagged", "According to the principle of proportional sampling:", "[\n\frac{200}{N} = \frac{36}{180}\n]", "### Simplifying and Solving for N", "First, simplify the right-hand side:", "[\n\frac{36}{180} = 0.2\n]", "So the equation becomes:", "[\n\frac{200}{N} = 0.2\n]", "Now solve for ( N ):", "[\nN = \frac{200}{0.2} = 1000\n]", "### Interpretation", "The estimated total bee population ( N ) is 1,000. This means, based on the sample data, that there are approximately 1,000 bees in the studied habitat. The accuracy of this estimate depends on whether the sample is representative and how randomly tagged bees mix within the population.", "### Why This Method Works", "Proportional sampling assumes that:\n- Tagged bees behave similarly to untagged bees (no altered behavior due to tagging)\n- The population is closed (no significant immigration or emigration during the study)\n- All bees have an equal chance of being captured in the sample", "These assumptions are crucial—violations can introduce bias.", "### Practical Applications", "For researchers and conservationists, this method provides a rapid, cost-effective way to estimate bee abundance without counting every individual. It is especially useful in monitoring endangered pollinators or assessing the impact of environmental changes on bee communities.", "---", "Conclusion", "Using setup ( \frac{200}{N} = \frac{36}{180} ), we find the total bee population is 1,000. This example illustrates how simple yet powerful mark-recapture sampling is in ecological research—turning a small, representative sample into a reliable estimate for large populations. By applying proportional sampling carefully, scientists can monitor bee populations effectively and support vital conservation strategies."]
