Fair price of a sum-divisible-by-three bet

Two fair dice are rolled. A bet pays $10 if their sum is divisible by 3 and $0 otherwise. What is its fair price (in dollars, to two decimals)?

Show hints (2)+
  1. Winning sums are 3,6,9,123,6,9,12 - but weight each by how many of the 3636 rolls produce it, not by counting sum-values.
  2. 2+5+4+1=122+5+4+1=12 rolls, so P=1236=13P=\tfrac{12}{36}=\tfrac13; price =13×10=\tfrac13\times 10.

Answer

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3.33 (± 0.01)

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Asked at: Timed Mental-Math & Sequences, Game-Based Aptitude

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