I am interested in finding the probability that on any single roll of dice you can have an exact number of successes. However, depending on the particular dice success is a different probability and I can mix dice types. Example for colors of six sided dice:
- Green Dice
- 6/6 sides are success
- Yellow Dice
- 4/6 Sides are success
- White Dice
- 2/6 sides are success
Looking around its easy to find information for rolling all successes with the dice being independent (like Yahtzee with 5 dice all rolling a 1) can be referenced as:
1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6 = 1/7776
So, if I was to roll one of each color and wanted to know what the probability of rolling 2 successes is rather than all, could I reference it as something like this but do chance to fail on later dice possibly? Example of my thinking:
6/6 * 4/6 * 4/6 (white fail chance) = 96/216 = 4/9
The order of the dice rolling for the problem doesn't matter aas they all go at once, but if I reverse this for white to be sucess and yellow to fail it becomes
6/6 * 2/6 * 2/6 (yellow fail chance) = 24/216 = 1/9
So by ordering it like this I have given the value weight which I do not intend, but the different probability's complicating this makes me thing that I am going about it the wrong way or missing something else. Mathematics has never been my best subject even if it is interesting. The order would matter with the green dice though since if I rolled 2 of them I could never get only 1 success because there is a 100% probability to get 2 successes.
Is there a better way to look at this or compensate for the fact that the order of success doesn't matter only the number of success?