Question: A soil scientist at the USDA selects 4 soil samples at random from a field containing 6 nitrogen-rich and 5 carbon-rich plots. What is the probability that exactly two of the selected plots are nitrogen-rich? - Treasure Valley Movers

Why Random Soil Sampling Matters—and How Probability Shapes Agriculture

In an era where precision defines modern farming, minor patterns can reveal big insights. A USDA scientist randomly selecting soil samples from a field offers more than a routine task—it illustrates a foundational principle of data science: estimating likelihoods to guide real-world decisions. Understanding how to calculate such probabilities isn’t just academic—it’s central to sustainable farming, policy planning, and future climate resilience.

Last month, a growing interest emerged around how decision-makers across agriculture analyze soil health through statistical modeling. One compelling scenario centers on a USDA scientist selecting 4 soil samples from a controlled field with 6 nitrogen-rich and 5 carbon-rich plots. The question: What’s the probability exactly two of the selected samples are nitrogen-rich? This elementary probability problem reflects real-world challenges in sampling design, resource allocation, and trend forecasting—especially critical amid heightened focus on soil carbon sequestration and regenerative agriculture.

This isn’t just a math exercise. Farmers, researchers, and policymakers increasingly rely on such probabilities to predict how soil composition affects crop yields, water retention, and carbon capture. With soil health recognized as key to food security, the ability to estimate sampling outcomes empowers informed action without overreach.

Understanding the Probability Calculation

To find the chance exactly two of the four selected plots are nitrogen-rich, we apply combinatorics—combining expectations of chance and balance in random selection.

We start by defining the field’s composition: 6 nitrogen-rich (successes) and 5 carbon-rich (controls), totaling 11 plots. The sample size is 4, and “exactly two nitrogen-rich” means 2 from the nitrogen group and 2 from carbon-rich.

The number of ways to choose 2 nitrogen-rich plots from 6 is calculated as:
 C(6, 2) = 6! / (2! × (6–2)!) = (6 × 5) / (2 × 1) = 15

The number of ways to choose 2 carbon-rich plots from 5 is:
 C(5, 2) = 5! / (2! × 3!) = (