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Sociopath

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I was wondering if anyone might want to help me with some less common dice probability. The dice mechanic is similar to the board game Risk (two sets of dice being compared), but the dice pools are varying numbers of 10-sided dice on either side (2 pools compared of a varying number).

I'm having a little trouble wrapping my head around how to get statistical probabilities of who wins on either side of the dice roll.

My needs involve finding the probabilities of the chances of winning on either side of the equation. To win, a die must be higher than it's counterpart from the other dice pool, and dice are compared highest to highest down the line.

For instance, if I roll one pool of 3 10-sided dice against a pool of 4 10-sided dice, I receive 8, 7, and 6 from pool one and 8, 5, 6 in the second pool. The highest are compared to the highest (8 = 8, 7 > 6, 6 > 5). Equal pairs cancel each other out, so pool one wins with 2 hits (2 of his dice exceeded the remaining two of the opponent).

If I recall correctly, opposing pools of equal dice have about a 50% probability of winning and a 50% probability of losing (the chances for any result on either side are the same across the board). What I don't know how to calculate is the probability of unequal dice pools in win/lose terms.

Values for the following is what I'm trying to figure out:

Probability of 1 die pool vs. 5 dice pool, probability of the 1 die at least cancelling out a hit from 5 dice (the 1 die exceeding all 5 dice, at least negating one hit).

Probability of uneven pairs...

- 1d vs 2d

- 1d vs 3d

- 1d vs 4d

- 1d vs 5d

- 2d vs 3d

- 2d vs 4d

- 2d vs 5d

- 3d vs 4d

- 3d vs 5d

- 4d vs 5d

...with win chances vs. bigger and smaller dice pools.

Some special rules (if anyone takes this up), a roll of 1 in any pool is discarded automatically; they don't exceed any die, and, being the poorest result possible, is discarded even if it does not have a matching die. So, any roll of 1 is automatically thrown out and not considered - except for the purposes of losing a die in your pool if it shows up.

I was thinking of doing probability of a theoretical 9-sided die, but I quickly threw that idea out because it would skew the results; using a 9-sided die would result in probability for the dice always returning higher than 1, which would not be the case.

I'd really appreciate help on this matter. I'm decent at math, but probability and statistics is a bit beyond my normal day-to-day.