Intro
I am currently developing a board-game and am trying to balance things around my dice-mechanic, but for that I need the probabilities of the outcomes and after countless fruitless hours of calculation I am turning to this community.
I apologize for improper grammar and technical terms, since English is not my mother tongue.
The Mechanic
A player has X (usually between 1 and 10) number of six-sided-dice which are labeled:
- three times: 1
- two times: 2
- one time: 4
After the player throws the dice, they group all ones together in one pool, all twos together in another pool and all fours together in yet another pool. They then select one of those pools, to make a play in the game.
Relevant is here, that the value of a pool is determined by the sum of all its faces, als well as the total number of dice in the pool. Higher is better. So in some cases it is more relevant to pick the pool with the highest summ, while sometimes the pool with the higher number of total dice is more helpful.
An Example
I throw 5 dice and the come up: 1, 1, 1, 2, 4. So I group them up in Pool_1={1,1,1}, Pool_2={2} and Pool_4={4}.
Pool_4 has the highest sum with 1x4=4, but Pool_1 has the highest number of dice with 3. Depending on my ingame situation I would choose one or the other, while Pool_2 in this case offers no benefit.
My Question
When I throw X dice, what is the probability of …
- reaching or exceeding the sum of A, when I pick the pool with the highest sum?
- reaching or exceeding the amount B of dice in a pool, when I pick the pool with the highest number of dice?
- reaching or exceeding the amount C of dice in a pool, when I pick the pool with the highest sum?
Outro
I would be very grateful if someone could help me solve this conundrum, so I could continue working on my project, to make it as fun and fair as possible.
Thank you!