No Stupid Questions

If your goal is to roll a 7, then no, weighting one die doesn't help, because it doesn't matter which side comes up in the weighted die.

(Another way to look at it is that you can place 1 die on whatever side you like, then roll the 2nd die: you still have a 1/6 chance of rolling a combined 7).

However that is only the case for a 7, because you can roll a 7 with any combination of the first die (and a particular value in the 2nd). If your goal is to roll a 12, then weighting one die towards 6 will affect the odds, because you need a 6 on that first die to roll a 12; any other outcome makes it impossible.

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.

You're marginally less likely to roll a 7 with a single weighted die.

My reasoning: if you use your weighted die, whoever you're gambling against might figure it out. Now they're pissed, they're gonna give you the beat down. But you're packing, and OH SHIT YOU DIDN'T MEAN TO HIT HIM IN THE HEAD YOU WERE AIMING FOR HIS LEG SHIT SHIT and now you're serving jail time for manslaughter and dice aren't allowed in jail, thus you won't be able to roll any 7s for a while.

There is statistically no difference if one is weighted because it's gone from 6/36 to 1/6.

Your question is already answered correctly, so I'm just chiming in with thoughts on a similar situation :)

If you weigh both dice, it gets interesting again.

The obvious is to make one die roll always 3 and the other always 4, and get 1/1 chance of 7 but that's boring and you'll only get a few throws in before you're obviously cheating.

Dice are arranged so that opposite sides always add up to 7, meaning you can get 1,2,3 around one corner and 4,5,6 around the opposite corner. So if you weigh opposite corners on each die, you get a 1/3 chance of rolling a 7 by varying combinatons. You might get away with a few more rolls like that.

If a die is weighted, the first roll is no longer 1/6 probability to get a 7; the roll isn’t random and there isn’t enough info. I think though it’s 3/12 (1/4 for a 7 (6+1, 5+2, 4+3)? Maybe not. I hate this shit.

Tell me the weights and I told you the probability of getting 7.

At the moment you have P(x) = {X=1 : 1/6, X=2: 1/6...X=6: 1/6). Give me the new P(X) and I calculate this for you.