{"knowledge_schema": {"broad_category": "Mathematics \u2192 Linear Algebra \u2192 Quantum Mechanics", "refinement": "This problem involves calculating probabilities related to quantum measurements using matrix operators and state vectors.", "specific_scope": "The focus is on determining the probability of obtaining a specific measurement outcome (1) from a given state vector and observable matrix.", "goal": "Calculate the probability of measuring the observable Q to yield the value 1 based on the provided state of the system."}, "summary": "This problem is rooted in linear algebra and quantum mechanics, where we calculate the probability of a measurement outcome based on a state vector and an observable represented by a matrix. The task involves applying the observable matrix to the state vector to find the probabilities of different measurement outcomes. Specifically, we need to compute the probability of measuring the observable Q to yield the value 1, which is determined through the inner product of the state vector with the corresponding eigenvector of the observable. The answer to this calculation is 1/4."}