Question: An ichthyologist is studying a population of fish in a lake and tags 10 fish. Later, a sample of 15 fish is randomly caught. What is the probability that exactly 4 of the tagged fish are in the sample, assuming the lake contains 200 fish total? - Coaching Toolbox

Understanding the Context

Across conservation circles and data-driven public engagement platforms, this type of problem reflects growing interest in ecological monitoring and sustainable resource management. With rising concerns over biodiversity, water quality, and fish population trends, understanding how sample sampling works adds clarity. More people are drawn to data-backed stories that combine on-the-ground science with statistical storytelling. This question sits at the crossroads—where ecology meets interpretation—and resonates with audiences curious about real-world applications of probability, not just abstract math.

How These Fisheries Studies Actually Work

Imagine a lake home to 200 fish, 10 of which have been carefully tagged for research. Scientists catch a random sample of 15 fish to estimate population size, monitor health, or assess migration patterns. The core question is: What’s the chance exactly 4 tagged fish appear in this smaller group? To solve this, scholars rely on a statistical model known as the hypergeometric distribution. Unlike simple ratios, this method accounts for sampling without replacement in a finite population, making predictions more accurate than chance-based guesses.

This approach reflects real challenges faced by fisheries biologists: sampling from a closed or semi-closed population, aiming for precision despite limited data, and avoiding bias from random chance. When 10 tagged fish are present among 200, catching a 15-fish sample means each fish has a roughly equal chance of inclusion—critical for modeling that sets the foundation for meaningful conclusions.

Key Insights

Calculating the Odds: The Math Behind the Discovery

To find the probability of pulling exactly 4 tagged fish from the 15-catch, we use hypergeometric probability. This formula accounts for the number of tags, total fish, and sample size:

Let

• N = total fish (200)
• K = tagged fish (10)
• n = sample size (15)
• k = desired tagged fish in sample (4)

The probability is:

P(X = 4) = [C(10,4) × C(190,11)] / C(200,15)

Final Thoughts

Here’s what each piece means: