Problem 20 · 1995 AJHSME
Hard
Counting & Probability
symmetrycomplementary-counting
Diana and Apollo each roll a standard die, obtaining a number at random from 1 to 6. What is the probability that Diana's number is larger than Apollo's number?
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Answer: B — 5/12.
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Hint 1 of 2
The two players are interchangeable, so 'Diana's number is bigger' and 'Apollo's number is bigger' must be EXACTLY as likely as each other. Use that fairness instead of listing cases.
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Hint 2 of 2
Every outcome is one of three things: Diana bigger, Apollo bigger, or a tie. Knock out the ties first, then the symmetry splits what remains evenly in two.
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Approach: symmetry — peel off the ties, split the rest in half
- Key insight: the dice are identical and the two rolls are interchangeable, so 'Diana > Apollo' and 'Apollo > Diana' are equally likely. The only outcome that breaks the symmetry is a tie — so handle ties separately.
- Total outcomes: 6 × 6 = 36. Ties (1-1, 2-2, …, 6-6) are 6 of them, leaving 36 − 6 = 30 where someone is strictly bigger.
- By the symmetry, half of those 30 favor Diana: 15 outcomes. Probability = 1536 = 5/12.
- Why this transfers: whenever two players or choices are interchangeable, the 'one beats the other' cases are equal — so P(A wins) = (1 − P(tie)) ÷ 2. Find the ties, and the rest is free. Trap: ½ forgets the ties; 5/12 is just under ½, which makes sense.
Another way — direct count:
- Count outcomes where Diana > Apollo by Apollo's roll: if Apollo rolls 1, Diana wins with 2–6 (5 ways); rolls 2 → 4 ways; … rolls 5 → 1 way; rolls 6 → 0 ways.
- Total favorable = 5 + 4 + 3 + 2 + 1 + 0 = 15, out of 36, giving 5/12 — matching the symmetry answer.
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