A science policy analyst is reviewing the distribution of grants for three categories of projects: renewable energy, AI research, and public health. There are 5 grants for renewable energy, 7 for AI research, and 3 for public health. If a committee randomly selects 3 grants to review, what is the probability that exactly 2 are for AI research and 1 is for public health?

["Science Policy Analyst Examines Grant Distribution: Probability of Selecting Exactly 2 AI Research and 1 Public Health Grant", "In the evolving landscape of scientific advancement, government and institutional funding plays a pivotal role in shaping innovation across key sectors. A recent analysis by a science policy analyst focused on grant distribution across three critical areas: renewable energy, artificial intelligence research, and public health. With 5 renewable energy grants, 7 AI research grants, and 3 public health grants available, the unit evaluated the likelihood of a randomized selection process favoring specific categories.", "Specifically, the committee plans to randomly select 3 grants out of the total 15 (5 + 7 + 3) for detailed review. The analyst’s investigation queried: What is the probability that, among these three selected grants, exactly two are from AI research and one from public health?", "---", "### Understanding the Selection Scenario", "To compute this probability, we apply principles of combinatorics and probability theory. The total number of ways to choose any 3 grants from 15 is given by the combination formula:", "[\n\binom{15}{3} = \frac{15!}{3!(15-3)!} = \frac{15 \ imes 14 \ imes 13}{3 \ imes 2 \ imes 1} = 455\n]", "This represents the total possible review panels the committee might assess.", "Next, we calculate the number of favorable outcomes—selecting exactly 2 AI research grants and 1 public health grant, with no AI or public health grants included in the selection other than as specified.", "- Number of AI research grants: 7 → choose 2 → (\binom{7}{2} = \frac{7 \ imes 6}{2} = 21)\n- Number of public health grants: 3 → choose 1 → (\binom{3}{1} = 3)", "Since the grants from renewable energy must not be selected, only the 7 + 3 = 10 non-AI, non-public health grants remain (5 renewable energy + 5 remaining public health? Wait—no: total public health is 3, AI is 7, so renewable energy is 15 – 7 – 3 = 5). So, excluding AI and public health, there are 5 renewable energy grants.", "But in selecting 3 grants with exactly 2 AI and 1 public health, renewable energy grants are not included at all in the favorable outcome.", "Hence, the number of favorable combinations is:", "[\n\binom{7}{2} \ imes \binom{3}{1} \ imes \binom{5}{0} = 21 \ imes 3 \ imes 1 = 63\n]", "Note: Since only 3 grants are selected, and we want no renewable energy grants included (and exactly 2 from AI, 1 from public health), we must pick all 3 from the 7 AI and 3 public health—specifically, 2 from AI and 1 from public health.", "---", "### Calculating the Probability", "The probability is the ratio of favorable outcomes to total possible outcomes:", "[\nP = \frac{\ ext{Favorable outcomes}}{\ ext{Total outcomes}} = \frac{63}{455}\n]", "Simplify the fraction:", "Divide numerator and denominator by 7:", "[\n\frac{63 \div 7}{455 \div 7} = \frac{9}{65}\n]", "---", "### Conclusion", "The probability that a random selection of 3 grants includes exactly 2 from AI research and 1 from public health is (\frac{9}{65}), or approximately 0.1385 (13.85%). This analysis underscores the importance of strategic grant allocation and transparent review processes in advancing science policy across vital domains.", "By quantifying selection likelihoods, science policy analysts can ensure fairness, maximize impact, and support evidence-based decisions in funding distribution. This approach helps maintain public trust and aligns resources with national innovation priorities.", "---", "Keywords: science policy analyst, grant distribution, probability calculation, AI research funding, public health grants, renewable energy grants, statistical analysis, funding allocation, scientific grants, committee selection probability."]