Probability of rolling X successes with differently sized dice I'm trying out different systems for a board game, and my (weak) probability skills are failing me.
The basic mechanic is as follows: you roll a number of six-sided dice (d6). Any die showing 5 or more is a success.
A specific bonus can "upgrade" part of your pool to a larger die (let's say a 10-sided dice, or d10).
So for example:
regular roll: roll 7d6; any die showing 5 or more is one success.
same roll with bonus: roll 4d6 and 3d10; any die showing 5 or more is one success.
I'd like to learn how to calculate the chance of getting X successes with the mixed pool (either "X" successes or "X or more" successes).
this thread thaught me how to find the probability for the homogenous pool, but I have no idea how to change the formula to accomodate the differently-sized dice.
I would like to learn the general formula, to be adjusted for different size of dice (such as d8 and d12) and for a larger or smaller pool of mixed dice as well.
Thanks in advance, sorry for any English mistake I may have made.

I jump directly to the second question: 

same roll with bonus: roll $4d6$ and $3d10$; any die showing 5 or more
  is one success.

4d6: The probability that one die show a 5 or a 6 is $\frac26=\frac13$. Now you can use the binomial distribution to calculate that x dice out of $4$ have an success.
$$P(X=x)=\binom{4}{x} \left( \frac{1}{3} \right)^x\cdot \left( \frac{2}{3} \right)^{4-x}$$ 
Similar for $3d10$
$$P(Y=y)=\binom{3}{y} \left( \frac{6}{10} \right)^y\cdot \left( \frac{4}{10} \right)^{3-y}$$ 
Let´s  denote $s$ as the number successes. We have to calculate the sum of the successes and fails to get the probability for $S=x+y$ successes and $3+4-x-y=7-s$ fails.  At maximum we can have $7$ successes. Thus
$P(S\geq s)=P(S=s)+P(S=s+1)+\ldots+P(S=7)$
As an example I show how to calculate $P(S=5)$. For this purpose I made a table for the combination of the successes for $x$ and $y$
$\begin{array}{|c|c|}\hline x&y \\ \hline 2&3 \\ \hline 3&2 \\ \hline 4&1 \\ \hline\end{array}$
Note that $x=1$ is not possible since $y\leq 3$. This shows why there is no "simple" formula. The random variables  $X$ and $Y$ are independent. Thus $P(X\cap Y)=P(X)\cdot P(Y)$. The single probabilities are 
$$P(X=2,Y=3)=\binom{4}{2} \cdot \left( \frac{1}{3} \right)^2\cdot \left( \frac{2}{3} \right)^{4-2}\cdot \binom{3}{3} \cdot  \left( \frac{6}{10} \right)^3\cdot \left( \frac{4}{10} \right)^{3-3}$$
$$P(X=3,Y=2)=\binom{4}{3} \cdot \left( \frac{1}{3} \right)^3\cdot \left( \frac{2}{3} \right)^{4-3}\cdot \binom{3}{2} \cdot  \left( \frac{6}{10} \right)^2\cdot \left( \frac{4}{10} \right)^{3-2}$$
$$P(X=4,Y=1)=\binom{4}{4} \cdot \left( \frac{1}{3} \right)^4\cdot \left( \frac{2}{3} \right)^{4-4}\cdot \binom{3}{1} \cdot  \left( \frac{6}{10} \right)^1\cdot \left( \frac{4}{10} \right)^{3-1}$$
Finally the single probabilities have to be summed up to get $P(S=5)$.
