*astragaloi*. They survive to this day in the children’s game known as Jacks. However, knucklebones were also used as dice.

A) Because of their peculiar shape, the bones only had four numbered faces. Traditionally, these had the values 1, 3, 4 and 6. The approximate probabilities of rolling each number were in the ratios 1 : 3 : 3 : 1. Playing with 3 such knucklebones, and assuming these probabilities are exact, what is the probability of rolling a sum total of 8?

B) From what we know of the ancient Greeks and Romans, they tended to play with 4 knucklebones and particular throws had distinctive names. A Venus (or Aphrodite) was a roll in which all four different numbers appeared, (1, 3, 4, 6); a Dog was the lowly (1, 1, 1, 1)! What is the probability of rolling a Venus?

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