2Question: A food scientist is analyzing a batch of 120 samples of a new plant-based protein. If the scientist randomly selects 5 samples for taste testing, what is the probability that exactly 2 of them are from the first 30 samples produced?

["Understanding the Probability Behind Taste Testing: A 2Question Analysis on Plant-Based Protein Samples", "When developing innovative food products, precision in testing is crucial. Take, for example, a recent experiment by a food scientist analyzing 120 batches of a new plant-based protein. After production, the scientist randomly selects 5 samples for taste testing. A common question arises: What is the probability that exactly 2 of these 5 selected samples come from the first 30 units produced?", "This article breaks down the probability calculation using combinatorics, explains the underlying statistical method, and highlights why such analysis matters in food science research and quality control.", "---", "### Step-by-Step: A Combinatorial Approach", "To determine the probability of selecting exactly 2 samples from the first 30, we apply the concept of hypergeometric distribution. This distribution models outcomes in scenarios involving sampling without replacement from a finite population.", "#### Given:\n- Total samples (population size): ( N = 120 )\n- Samples selected (sample size): ( n = 5 )\n- Favorable initial samples: 30 out of 120\n- Desired samples from initial group: 2", "This setup fits the hypergeometric model because:\n- The selection is random and done without replacement.\n- We track counts in “successes” (first 30 samples) and “failures” (remaining 90 samples).", "---", "### Hypergeometric Probability Formula", "The hypergeometric probability is calculated as:", "[\nP(X = k) = \frac{{\binom{K}{k} \cdot \binom{N-K}{n-k}}}{\binom{N}{n}}\n]", "Where:\n- ( N = 120 ): total samples\n- ( K = 30 ): samples from the first 30 units\n- ( n = 5 ): total samples selected\n- ( k = 2 ): exactly 2 from the first 30\n- ( \binom{a}{b} ) is the binomial coefficient, representing combinations: ways to choose ( b ) from ( a )", "---", "### Applying the Values", "[\nP(X = 2) = \frac{{\binom{30}{2} \cdot \binom{90}{3}}}{\binom{120}{5}}\n]", "Now compute each component:", "- ( \binom{30}{2} = \frac{30 \ imes 29}{2} = 435 )\n- ( \binom{90}{3} = \frac{90 \ imes 89 \ imes 88}{6} = 117480 )\n- ( \binom{120}{5} = \frac{120 \ imes 119 \ imes 118 \ imes 117 \ imes 116}{120} = 190,578,024 )", "Plug in:", "[\nP(X = 2) = \frac{435 \ imes 117480}{190,578,024} = \frac{51,073,800}{190,578,024} \approx 0.2684\n]", "So, the probability is approximately 26.84%.", "---", "### Why This Matters in Plant-Based Protein Development", "Random sampling with correct probability ensures reliable taste profiling, critical for refining formulations and confirming consistency in mass production. Statistical rigor helps scientists avoid bias, validate quality, and accelerate product development with data-backed confidence.", "---", "### Summary", "- The probability of selecting exactly 2 samples from the first 30 out of 5 sampled follows a hypergeometric model.\n- Using combinations, we computed the exact probability: ~26.84%.\n- This method underpins quality assurance in food science, ensuring palatability testing reflects true product composition.", "By mastering such probability models, scientists can design smarter experiments, improve product safety, and enhance consumer satisfaction in the booming plant-based food industry.", "---", "Key Terms: hypergeometric distribution, combinatorics in food science, probability sampling, plant-based protein testing, statistical analysis, binomial coefficient, food quality control.", "For more insights on probability in food science, explore advanced statistical methods and sampling techniques in our full guide on scientific validation in protein development."]