Problem 18 · 2011 AMC 8
Medium
Counting & Probability
symmetryprobability
A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?
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Answer: D — 7/12.
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Hint 1 of 2
≥ means ‘greater OR equal,’ so handle the tie separately. The 6 tie outcomes are easy; then use symmetry for the rest.
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Hint 2 of 2
‘First > second’ and ‘first < second’ are mirror images — relabel the two dice and one becomes the other — so they have equal probability and split the non-tie outcomes evenly.
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Approach: use the symmetry between ‘greater’ and ‘less,’ handle ties separately
- Ties: 36 equally likely (first, second) pairs, and 6 of them are doubles, so P(equal) = 6/36 = 1/6.
- The remaining 30 non-tie outcomes split evenly between ‘first > second’ and ‘first < second’ (swapping the dice turns one into the other), so each is 15/36 = 5/12.
- We want ≥, which is ‘greater’ plus ‘equal’: 5/12 + 1/6 = 5/12 + 2/12 = 7/12.
- Worth keeping: when two events are mirror images, they share a probability — so you only compute the ‘extra’ (the ties) and split the rest.
Another way — count favorable cells in the 6×6 grid:
- For first = 1 there's 1 winning second (just 1); for first = 2 there are 2; … up to first = 6 giving 6.
- Total favorable = 1+2+3+4+5+6 = 21, out of 36, so 21/36 = 7/12.
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