URI CSC 340 F22 HW1 Q9

Base case:
If 1 distinct die is rolled, the number of outcomes where the sum of faces is an even integer equals the number of outcomes with an odd sum. This can be proven by enumerating the possibilities, all with equal probability:
even: 2, 4, 6
odd: 1, 3, 5

Inductive hypothesis: If n distinct dice are rolled, the number of outcomes where the sum of faces is an even integer equals the number of outcomes with an odd sum ⇒ If n + 1 distinct dice are rolled, the number of outcomes where the sum of faces is an even integer equals the number of outcomes with an odd sum. This can be proven by enumerating the possibilities, all with equal probabilities:
even: even + 2, even + 4, even + 6, odd + 1, odd + 3, odd + 5
odd: even + 1, even + 3, even + 5, odd + 2, odd + 4, odd + 6

∴ If n distinct dice are rolled, the number of outcomes where the sum of faces is an even integer equals the number of outcomes with an odd sum ∀n ≥ 1