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The article focuses on the solution to matching hats problem using mathematical probability. It is stated that the solution is a good practice with indicator random variables, which takes the value of one when an event has occurred. The said indicator random variables have a special property in which their expected value is simply the probability of the event that occurs. Moreover, the mathematical solution in finding the probability of the given cap that has been matched with its owner is further discussed.

We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mired quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly non-Abelian generating group, and show that if the generator satisfies a weighted norm constraint, then the LSM is optimal. In particular, for pure-state GU ensembles, the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets, the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time. [ABSTRACT FROM AUTHOR]