Professor Rando has 4 grad students, Daphne, Max, Mindy, and Sam. The professor proposes the following game: he picks two integers independently and uniformly at random from 1 to 5 (inclusive), so all 25 ordered outcomes are equally likely. He then privately tells each of his students a different fact about the two numbers. He tells:
- Daphne the absolute difference of the integers,
- Max the maximum of the integers,
- Mindy the minimum of the integers, and
- Sam the sum of the integers.
Then, each day until the game ends, he gathers his students and asks them one at a time, always in alphabetical order (Daphne, then Max, then Mindy, then Sam), to name the two integers. "Naming the integers" means naming the unordered pair (for example, "2 and 5"); nobody has to say which number was drawn first.
Each student gets one chance per day, when called upon, and this repeats every day until someone answers. A student answers only if he or she knows the pair for certain; otherwise that student stays silent for that day. Everyone hears who stays silent (including students called earlier the same day), everyone knows which kind of fact (difference, maximum, minimum, sum) each student was told, and everyone knows these rules; all students are perfect logicians, and there is no collusion. The first student to answer (always correctly) wins, and the game ends. The professor draws his numbers once, at the start, not each day.
What is the probability that each of Daphne, Max, Mindy, and Sam wins?
Give the four winning probabilities as exact fractions, separated by commas, in the order Daphne, Max, Mindy, Sam.