I'm rusty on probability and, even so, I don't think I know how to deal with this situation. Here's the problem.
Let's say I have a cup of $n$ dice. Each dice face has either a shotgun, a brain, or a runner on it (images on the dice.) There are three types of dice: red, green, and yellow. The red dice have 3 shotguns. The greens dice have 1 shotgun. The yellow dice have 2 shotguns.
There are 13 dice total, and dice that on the table in front of me have been drawn from the cup. The player rolls 3 dice at a time. There are 3 red dice, 6 green dice, and 4 yellow dice.
Let's say I have one of the dice in front of me, and the rest are in the cup. I know the colour of the dice in front of me. I have to draw 2 more dice from the cup at random. I won't know what colour they are.
I then roll all three dice.
What are the chances that I roll 2 or more shotguns?
There are other situations in the game where I have 0-2 dice in front of me, and I would want to know the chance of rolling 1-3 shotguns. If someone can explain the calculation to me, that would be great.
I can calculate for when I'm rolling 3 dice of known colour, but when I'm drawing from the cup, I can't figure it out.