RE: The Paradox of Non-Transitive Dice and Our Imperfect Intuitions
So, if I understood what your post said, the probabilities of each die you provide winning when pitted against one other were based on one roll of each competing die.
But if the two dice are rolled more than once against each other, it gives more chances for the die with a favorable 1/6 option to actually hit it, though it also gives the one with the 5/6 option that many more chances of winning.
Also, since I'm not a dice expert, does the probability change at all based on how you hold the die, or how you roll it each time?
In other words, while the probability of winning against another die could be set based on amounts and their availability, the person rolling the die could be adept enough to manipulate the outcomes by figuring out beforehand what starting position and what potential force and angle might result in more favorable odds.
It's not really one roll, I'm examining the probability for each possible roll. The probabilities give you the expected distribution of win in a long series of rolls, not just a single roll. It's just that each face or roll of a certain die has a specific probability to come up and that allows us to calculate the total probability for each outcome. In fact, you could draw a six by six table with the faces of one die for rows and the faces if the other for the columns. The cells of that table would be the possible outcome of the two dice being rolled. Each cell of the table has an equal probability of happening. If you do that table for the yellow vs blue scenario, you are going to get 21 out of the 36 cells which is the same as the 7/12 probability and the blue is going to get 15 out of the 36 cells which is the same as the 5/12 probability.
It's not what the post examines. It examines the odds that the die itself affords you with fair random rolls. I'm just examining the dice themselves. If you have a player with some skill, they would still do better with the yellow dice than with the blue dice. The yellow dice still has the same intrinsic advantage, just the player can get other ones as well. If we are talking about a proper casino situation where the dice need to bounce off of an unevenly shaped backboard, than all claims of skill tend to be scams to get people to pay you for lessons, confirmation bias, or both. But that's a whole other story and it's really beside the point here.
Also keep in mind that casinos are making tons of money with the odds being in their favor with just a few percentage points while those dice afford you significantly higher advantages. So if it's fair to say that casinos are rigged to win, than those dice are even more unbalance and rigged even harder if that makes sense.