An anthropologist studies 8 remote communities, each with a different ceremonial dance performed an average of 3 times per year. She selects 3 communities at random to conduct in-depth ethnographic fieldwork. Let X be the random variable representing the total number of dances observed across the 3 selected communities in one year. What is the expected value of X?

["Understanding Ceremonial Rhythms: A Study in Ethnographic Patterns", "In the pursuit of understanding human cultural expression, an anthropologist embarks on a groundbreaking study across eight remote communities, each practicing a unique ceremonial dance with remarkable consistency—each dance performed an average of three times per year. Beyond mere observation, this in-depth ethnographic research focuses on quantifying cultural frequency: how often do these sacred performances occur across selected communities?", "Let’s define the key variables. With 8 communities each hosting a traditional dance at an average rate of 3 times annually, the total annual dance incidents across all communities amount to:", "[\n8 \ ext{ communities} \ imes 3 \ ext{ dances/year} = 24 \ ext{ dances/year}\n]", "But when the anthropologist focuses on only 3 randomly selected communities, we shift from population-level averages to a random variable representing total dances observed in one year in this subset.", "Let ( X ) be the random variable representing the total number of dances recorded across the 3 randomly chosen communities.", "Because the selection is random and the average number of dances per community is constant, ( X ) follows a hypergeometric-like additive structure—though due to uniformity in mean output, we can simplify expectations without complex concentration modeling.", "Why a hypergeometric model? In idealized random sampling with replacement of community performance frequency, each community "contributes" an expected number of dances proportional to its place in the full set. However, here the underlying process is deterministic average under random selection—so we compute the expected value directly via linearity of expectation, treating each community’s contribution as a Bernoulli-type indicator scaled by average frequency.", "Each of the 8 communities contributes, on average, 3 dances per year. When selecting 3 communities at random without regard to order, the expected number of dances observed in a year is simply:", "[\nE(X) = 3 \ ext{ (average per community)} \ imes 3 \ ext{ (selected communities)} = 9\n]", "This holds because expectation is linear, and even though the communities vary in identity and dance typology, the average annual dance frequency per community is consistent. Random selection does not alter the expected total—it preserves the average over the sample.", "Thus, the expected value of the total number of dances observed across the 3 selected communities in one year is:", "[\n\boxed{9}\n]", "This result underscores how statistical modeling enhances anthropological analysis—not by reducing cultural richness, but by illuminating predictable rhythms behind ceremonial life. Whether tracking ritual frequency or social cohesion cycles, such quantitative insights empower deeper interpretation of human traditions across time and space."]