Its 1/6 to roll a 7 with two 6-sided dice. You can get weighted dice that make it more likely to land on a certain number.
Does having one weighted dice change the odds at all? My gut says no but reality is a tricky bitch and I’m convinced im wrong somehow.
You’re right! Let’s say we have two dice:
D₁ is fair and has a 1/6 probability of rolling each number from 1-6.
D₂ is weighted, with probabilities P₁, P₂, P₃, P₄, P₅, P₆ to roll each number.
We roll D₁, and get a number with the following probability distribution:
1: 1/6
2: 1/6
3: 1/6
4: 1/6
5: 1/6
6: 1/6
We roll D₂, and get a number with the following probability distribution:
1: P₁
2: P₂
3: P₃
4: P₄
5: P₅
6: P₆
We find the probabilities of every combination of rolls that yields a 7:
1+6: 1/6 P₁
2+5: 1/6 P₂
3+4: 1/6 P₃
4+3: 1/6 P₄
5+2: 1/6 P₅
6+1: 1/6 P₆
Adding these together to get the total probability of rolling a 7, we get 1/6 (P₁ + P₂ + P₃ + P₄ + P₅ + P₆). Since the probabilities of rolling each number must sum to 1, we get a probability of 1/6 to roll a 7, and your gut is right. :)
7 is the only number where this property holds. Other numbers will have a probability dependent on the weighting of the die, which could be calculated with a similar method.