I'm looking for some good formulas for my dice.

I am currently having players roll 4 6-sided dice up to 3 times, saving or not saving any dice they roll until the last roll, where they must keep it. (fairly standard Yahtzee style)

Lets say I have one side that is very bad, and one side that is very good.

if the player is selecting for or against the dice, what is the likely average number of good/bad icons rolled?

I know the likely hood of the specific number you want never showing up is (5 misses/6 sides)^12 rolls total = 11.2%, thus the likelihood of seeing it at least once is 100%-11.2% = 88.8

and the likely hood of getting all the good specific side on the first roll is (1/6)^4 = 0.07%

but how do I calculate the in-between likelihoods? the average? preferably as a formula that I can alter the number of dice and rolls allowed until I can tune it.

Thanks for the help!

I tried a table method

Table:

https://docs.google.com/spreadsheets/d/1JezJXtFfTZ2qurpfmQkqG41_F5LAqRXg...

I broke down all the possible ways to get 4 good dice, 3 good, 2 good 1 good 0 good, and filled in with the chances from Anydice what the likelihood for each roll is.

But I must have a mistake somewhere! The total for all the chances for all the scenarios is only 55%, and it should be 100.

@X3M

How does my table look?

@anyone else

There has got to be a formula out there for something like this.

Thanks