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A hypergeometric distribution models the probability of obtaining a specific number of "successes" in a sample drawn without replacement from a finite population. In the fish tagging scenario, "success" means catching a previously tagged fish. Because the fish are selected without replacement, the hypergeometric distribution is perfect for this problem. Compared to the binomial distribution, the hypergeometric model adjusts for the fact that each fish selection alters the pool of available fish.
The probability of catching exactly 10 tagged fish out of 100 caught is captured by:

• Numerator: The product of the combination of selecting 10 tagged fish from the initially tagged and 90 untagged from the rest.
• Denominator: The combination of choosing 100 fish from the total population of trout.

Thus, it is a perfect way to capture the dynamics of probability in scenarios involving finite populations like our lake.

In statistics, the maximum likelihood estimation (MLE) is a method used to estimate the parameters of a statistical model. It finds the parameter values that maximize the likelihood function, meaning it tries to find the parameter setup that makes the observed data most probable. For this scenario, it means determining the most likely number of trout, given the tagging data.
Using MLE involves setting the probability function derivative to zero and solving for the population size. In step 4, this is done by comparing likelihoods of consecutive possibilities for total trout population number, modifying until the likelihood surface flattens. We find that adjusting the population size to maximize the probability of our observed catch data results in approximately 990 for the unknown quantity N, showcasing how the MLE technique pinpoints the optimal population estimation.

Proportion estimation is a statistical approach that infers an entire population's characteristics from a sampled subset. For the trout in our lake, the worker used the concept of sampling proportion. This means the observed portion of tagged fish in the second sample (10 out of 100) is assumed to mirror the entire lake's tagged population proportion.
By setting up the proportion equation \( \frac{100}{N} = 0.1 \), we estimate a total lake population \( N \). This type of estimation is advantageous because it is straightforward — only basic arithmetic operations and assumptions about uniformity in mixing are required. It doesn't require the complex calculations of alternative statistical methods, which is one reason it's powerful and commonly used in environmental studies and similar fields. With our data, it enabled estimating the lake contains roughly 1000 fish, highlighting how simple ratios can yield insights for population estimation.