I used three reddish, semi-transparent plastic dice with white dots (as I always did). My opponent used standard opaque, plastic ivory dice with black dots. I noticed nothing unusual about the dice and by the end of the run I was examining dice, cups, methods of rolling closely.
“Assume that a weighted die rolls the side that it favors with probability p, each of the sides adjacent to it with probability (1-p)/4, and never rolls the side opposite the favored side.”
This assumption does not match my recollection of the dice rolls. As I stated previously, I rolled 6′s, 5′s, 4′s, 3′s, 2′s, and 1′s. I also never rolled a 1,1,1 which should happen frequently if my dice were heavily weighed to roll 1′s. Nor do I remember rolling large numbers of 1′s.
Your probability model for a trick die also fails to match my observations of my opponents die rolls. E.g., in your model my opponent would be expected to roll similar numbers of 5′s, 4′s, 3′, and 2′s. However, he only rolled a 2 once and he rolled far more 5′s than 3′s.
Besides with your probability model for trick dice, I would have easily noticed if my opponent rolled a 6 84% of the time and I never rolled a 6 at all.
PS You used 26 in the above calculation. I had 26 armies and in Risk the attacker must have at least 4 armies to roll three attack dice. So the 3vs1 dice scenario only happened 23 times.
“all four dice were weighted”
I used three reddish, semi-transparent plastic dice with white dots (as I always did). My opponent used standard opaque, plastic ivory dice with black dots. I noticed nothing unusual about the dice and by the end of the run I was examining dice, cups, methods of rolling closely.
“Assume that a weighted die rolls the side that it favors with probability p, each of the sides adjacent to it with probability (1-p)/4, and never rolls the side opposite the favored side.”
This assumption does not match my recollection of the dice rolls. As I stated previously, I rolled 6′s, 5′s, 4′s, 3′s, 2′s, and 1′s. I also never rolled a 1,1,1 which should happen frequently if my dice were heavily weighed to roll 1′s. Nor do I remember rolling large numbers of 1′s.
Your probability model for a trick die also fails to match my observations of my opponents die rolls. E.g., in your model my opponent would be expected to roll similar numbers of 5′s, 4′s, 3′, and 2′s. However, he only rolled a 2 once and he rolled far more 5′s than 3′s.
Besides with your probability model for trick dice, I would have easily noticed if my opponent rolled a 6 84% of the time and I never rolled a 6 at all.
PS You used 26 in the above calculation. I had 26 armies and in Risk the attacker must have at least 4 armies to roll three attack dice. So the 3vs1 dice scenario only happened 23 times.