Probability of "at least amount" I have 4 six-sided dice.  One of them has values of [0,0,3,3,3,4], and the other three have values of [0,0,0,1,1,2].  The first dice has a 2/3 chance of rolling at least a 3.  If I roll the other three dice together, what are the odds of rolling at least a three?
I don't know what to call this, but that is the best way I know how to ask that question.  I have many dice like this in a game I play and I'd like to do some analysis.  Mostly six-sided dice with values ranging from zero to four, but there are some ten-sided dice with values ranging from zero to four, so a general formula would be great.  
So as a secondary question, I'd also like to solve that if I have a ten-sided dice with values [0,0,0,0,0,0,0,4,4,4], and three of the aforementioned [0,0,0,1,1,2] dice, what are the odds of rolling at least a sum of 5?
Thanks!
 A: We solve the problem of the probability of getting at least $3$ using three dice that have faces $(0,0,0,1,1,2)$. Mainly it is to illustrate technique. 
It is easier to find the probability $p$ of getting a sum less than $3$, that is, a sum of $0$, $1$, or $2$. Then the answer to your problem is $1-p$.
The following are all the ways we could end up with sum $\le 2$:
(i) all $0$;
(ii) two $0$ and a $1$;
(iii) two $0$ and a $2$;
(iv) one $0$ and two $1$;
We find the probability of each and add up.
(i) The probability is $\left(\frac{3}{6}\right)^2$.
(ii) The die that shows a $1$ can be chosen in $3$ ways. The probability it shows a $1$ is $\frac{2}{6}$. The probability the other two show $0$ is $\left(\frac{3}{6}\right)^2$. Thus we get probability $3\cdot \frac{2}{6}\cdot \left(\frac{3}{6}\right)^2$.
(iii) Essentially the same reasoning gives $3\cdot \frac{1}{6}\cdot \left(\frac{3}{6}\right)^2$.
(iv) The reasoning is the same as for (ii). We get $3\cdot \frac{3}{6}\cdot \left(\frac{2}{6}\right)^2$.
Add all the above probabilities to get $p$. Then as mentioned earlier the answer to your problem is $1-p$. 
Remark: If problems get much more complicated, and there are quite a few, simulation may be the way to go. 
