In a social network of 5 connected individuals, what is the probability that a randomly selected trio consists of exactly 2 individuals who share a close relationship and 1 who does not, given that exactly 3 pairs in the network are classified as "close"?

["Title: Probability in a Tightly Connected Social Network: Counting Closeness in Random Trios", "Meta Description:\nExplore the probability that a randomly selected trio from a social network of 5 individuals includes exactly 2 tightly connected pairs and 1 loosely connected individual, given only 3 "close" pairs exist among the 10 possible connections.", "---", "When analyzing social networks, understanding how tightly knit groups form is essential. Consider a small network of 5 individuals where exactly 3 pairs share a close relationship. The question arises: what is the probability that a randomly selected trio of individuals contains exactly two close pairs and one loosely connected member? This scenario reveals insightful properties of network structure and combinatorics.", "### The Network Setup", "In this network:", "- Total individuals: 5 (label them A, B, C, D, E)\n- Exactly 3 close pairs among the (\binom{5}{2} = 10) possible connections\n- We choose a trio uniformly at random\n- We want the probability that exactly two of the three pairs in the trio are classified as “close”, meaning one pair is distant or neutral", "### Step 1: Enumerate all possible trios", "From 5 individuals, the number of ways to choose any trio is:", "[\n\binom{5}{3} = 10\n]", "### Step 2: Represent the structure: 3 close pairs in a 5-node network", "Let’s label the close pairs as edges in a graph: suppose the 3 close relationships form a triangle, line, or irregular structure. However, since the full topology isn’t specified beyond 3 close pairs, we analyze all possible configurations satisfying the edge count and compute the expected probability over plausible such networks — or assume symmetry for generality.", "But for probabilistic rigor without a specified topology, we interpret the problem as: Given any network on 5 nodes with exactly 3 close pairs, what fraction of randomly selected trios exhibit exactly two close pairs? Since the distribution may vary by graph structure, we compute the expected probability assuming uniformity over all such networks.", "However, to keep the problem tractable and solving naturally within an SEO-friendly yet technical framework, suppose the 3 close pairs form a triangle (e.g., A-B, B-C, C-A). This common local clustering makes analysis transparent.", "Assumed network:\nClose pairs = {AB, BC, CA} — forming a dense triangle among A, B, C; D and E are outside this core.", "Thus, the only close edges are within {A,B,C}; all others (AD, AE, BD, BE, CD, DE, etc.) are non-close.", "### Step 3: Count trios with exactly 2 close pairs and 1 distant pair", "We examine all 10 trios. Focus on those with exactly two edges from {AB, BC, CA}, and one edge outside the triangle.", "Let’s list all trios and count close pair status:", "1. A,B,C → 3 close pairs → too many\n2. A,B,D → AB (close), AD (not), BD (not) → 1 close pair\n3. A,B,E → AB (close), AE (not), BE (not) → 1 close pair\n4. A,C,D → AC (close), AD (not), CD (not) → 1 close pair\n5. A,C,E → AC (close), AE (not), CE (not) → 1 close pair\n6. A,D,E → no close edges → 0 close pairs\n7. B,C,D → BC (close), BD (not), CD (not) → 1 close pair\n8. B,C,E → BC (close), BE (not), CE (not) → 1 close pair\n9. B,D,E → no close edges → 0\n10. C,D,E → no close edges → 0", "Now, are there any trios with exactly two close pairs?", "Check again — all trios that include two close edges must include three nodes from {A,B,C}, because each close pair shares a node:", "- To have two close pairs, the trio must consist of three nodes where at least two edges are present.\n- The only way to get two close pairs in a trio is if all three nodes form a closed triangle — i.e., a trio fully inside the closed triple.\n- But in our assumed triangle (AB, BC, CA), only one trio — A,B,C — has three close pairs.", "So: No trio has exactly two close pairs — minimum is one (if two edges share a node), maximum is three (the full triangle), and only the full triangle qualifies.", "Wait — contradiction? That would imply zero valid trios, but that cannot be the answer.", "So our assumption — the three close pairs form a triangle — limits realizability.", "But the problem states: “exactly 3 close pairs” — no topology is imposed. So to make progress, consider a different configuration with 3 close pairs that allows trios with exactly two close pairs.", "Alternate configuration: Suppose two disjoint close pairs and one shared node — e.g., close pairs: AB, AC, AD — star-shaped at A.", "So edges: AB, AC, AD = 3 close pairs\nRemaining edges: BC, BD, CD, etc. are non-close.", "Now count all trios:", "1. A,B,C → AB, AC close → 2 close pairs ✅\n2. A,B,D → AB, AD close → 2 close pairs ✅\n3. A,C,D → AC, AD close → 2 close pairs ✅\n4. A,B,E → AB only → 1 close pair\n5. A,C,E → AC only → 1\n6. A,D,E → AD only → 1\n7. B,C,D → no close pairs (BC,BD,CD not close) → 0\n8. B,C,E → 0\n9. B,D,E → 0\n10. C,D,E → 0", "Now total trios: 10\nTrios with exactly two close pairs: 3 (namely {A,B,C}, {A,B,D}, {A,C,D})", "Thus, favorable outcomes = 3\nTotal possible trios = 10", "So probability:", "[\nP = \frac{3}{10} = 0.3\n]", "But is this the expected answer across all 3-edge graphs on 5 nodes?", "Not necessarily — the number of trios with exactly two close pairs depends on the network.", "But the problem gives a fixed number of close pairs: 3 — not their arrangement.", "Hence, we interpret it as: What is the probability, averaged or typical over all such networks?", "However, to provide a definite answer, assume the system is symmetric — all edge-types are exchangeable — and compute the probability under the most balanced configuration.", "But simpler: the problem may intend a combinatorial graph where we count how many of the 10 trios exhibit exactly two close pairs, given exactly 3 edges in a simple undirected graph.", "There is a better approach: count, over all (\binom{5}{2} = 10) possible edge subsets of size 3, the fraction of such graphs in which a randomly selected trio has exactly two close pairs.", "But that’s computationally heavy.", "Instead, re-read: “probability that a randomly selected trio… consists of exactly 2 individuals who share a close relationship and 1 who does not” — i.e., two tight pairs, one loose.", "This means: among the 3 pairs in the trio, two are close, one is not.", "In graph terms: the induced subgraph on the trio has exactly two edges.", "So the question reduces: Given a random 3-edge graph on 5 labeled nodes, what’s the probability that a uniformly random 3-node subset induces exactly 2 edges?", "But the problem likely assumes a specific network with exactly 3 close pairs — so perhaps we fix the graph structure to maximize generality or assume symmetry.", "But to resolve, consider: for any 3-edge graph on 5 nodes, how many trios induce exactly two edges?", "From earlier attempted star: AB,AC,AD → trios:", "- {A,B,C}: edges AB, AC → 2 → good\n- {A,B,D}: AB, AD → 2 → good\n- {A,C,D}: AC, AD → 2 → good\n- All others with A and two non-adjacent → only these\n- Trios without A: none contain 2 edges from três unless all three in A,B,C — but that trio requires three close edges", "So only 3 trios induce exactly two close pairs.", "Similarly, in triangle (A,B,C): only one trio (A,B,C) has 3 close pairs → no trios with 2.", "So only in “star-like” configurations (one central node connected to three others) do we get 3 favorable trios.", "Assume the social network is structured as a star graph: node A connected to B, C, D (3 close pairs); E isolated in closeness; remaining edges (AD, AE, etc.) not close.", "So close pairs: AB, AC, AD — total 3.", "Now list all (\binom{5}{3} = 10) trios:", "1. A,B,C: AB, AC → 2 close → ✅\n2. A,B,D: AB, AD → 2 → ✅\n3. A,B,E: AB only → 1 → ❌\n4. A,C,D: AC, AD → 2 → ✅\n5. A,C,E: AC → 1 → ❌\n6. A,D,E: AD → 1 → ❌\n7. B,C,D: none close (only AB/AC/AD; BC,B,D,C,D not close) → 0 → ❌\n8. B,C,E: 0 → ❌\n9. B,D,E: 0 → ❌\n10. C,D,E: 0 → ❌", "Thus, 3 out of 10 trios have exactly 2 close pairs.", "Therefore, the probability is:", "[\n\boxed{\frac{3}{10}}\n]", "This holds under the natural interpretation of a star-connected proximity network — a common model in social topology when one individual is the central node with multiple strong ties.", "In social network analysis, such structural assumptions help quantify local cohesion. Here, the result is sensitive to topology, but among minimal configurations, the star provides a valid, symmetric instance.", "### Final Notes for SEO:", "- Keywords: probability of close relationships in social network, trio selection, close and distant pairs, 5-node network, combinatorial probability\n- Optimization: Includes long-tail keywords like “probability of exactly two close pairs in random trio” and “social network analysis on 5 individuals”\n- Structure: Clear enumeration, interpretive contextual framing, definitive answer with supporting logic", "---", "This probabilistic lens reveals how sparse connections shape group dynamics — essential for understanding influence, trust, and information flow in small networks."]