Find the Probability That One Shop Supplies the Top Two Coffees

Four different coffees receive scores, with two from shop A and two from shop B. What is the chance that both A coffees are scored above both B coffees? Give your counting argument.

Constraints

For this exercise, assume scores have no ties and the four coffees are exchangeable, so all 4! strict rankings are equally likely. Independent samples from the same continuous score distribution satisfy this assumption. Fair scoring alone should not be treated as proof that different shops have identical score distributions.

Clarifying Questions

• Can ties occur, and does 'better' require a strictly greater score?
• Is every strict ordering equally probable, or might the two shops have different score distributions?

Hint

Choose which positions belong to shop A You can count labeled rankings or count the two ranking positions occupied by A's coffees, as long as the sample space is consistent.

What a Strong Answer Covers

• The assumption that all rankings are equally likely, together with the event where A holds the first two positions.
• A favorable count divided by total count, done correctly and without counting any ranking twice.
• How the problem shifts if ties are allowed or if the shops' score distributions differ.

Follow-up Questions

• What is the probability that one shop—either A or B—takes both of the top two positions?
• How does the counting argument extend when A has m coffees and B has n coffees?

Overview

Calculate the probability that two coffees from one shop both outrank the two from the other shop, under a stated model with exchangeability and no ties.