Probability Both Drinks from One Coffee Shop Outscore Both Drinks from Another

You visit two coffee shops, A and B, and order two drinks at each. Each of the four drinks receives a quality score. What is the probability that both drinks from shop A have higher scores than both drinks from shop B? ```hint Only the order matters If all four scores come from the same continuous distribution, think about which features of the four scores the event actually depends on. ``` ### Clarifying Questions - Are the four scores independent draws from the same distribution, that is, are the two shops equally good? - Are the scores continuous, so that ties have probability zero, or can two drinks receive the same score? - Does "higher" mean strictly higher? ### What a Strong Answer Covers - The assumptions that make the question answerable (independence, identical distributions, no ties), stated explicitly - A symmetry argument over rankings of the four drinks, with a correct count of favorable rankings - A check of the result by a second method - Why multiplying pairwise or per-drink probabilities gives a wrong answer - How the answer changes with ties, unequal shops, or other numbers of drinks ### Follow-up Questions - Generalize: m drinks from A and n drinks from B. What is the probability that every A drink outscores every B drink? - What is the probability that the single best of the four drinks comes from shop A? - If scores are whole numbers from 1 to 10, so that ties are possible, how does the calculation change? - If shop A's drinks tend to be better, is the probability above or below the symmetric answer, and how would you compute it?