How to do this in your head:
Imagine you have a die with $a$ sides and one with $b$ sides ($b>a$), so the $a\times b$ combinations all have probability $\frac{1}{ab}$. If you wrote them out, they'd make an $a\times b$ table. You want the upper tail probabilities.
So if you wrote out your $a\times b$ table the largest outcome would be the bottom right entry, it would have value $a+b$ and probability $\frac{1}{ab}$. The second largest outcomes would be the two entries above and to the left of that one, with value one smaller, each with probability $\frac{1}{ab}$.
Each next smaller outcome would be on a diagonal above and to the left of the larger outcomes we considered before it. The count would increase until the top right corner of our table, where the outcome was $b+1$, which has $a$ entries at that value. (After we round the corner, the probabilities stay at $\frac{a}{a+b}$ until we get to a total of $a+1$, and then if we go back any earlier, we are going down the left arm of the trapezoid in the distribution as we head to the top left corner of the table.
The probabilities of the totals then look like this:
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... b b+1 a+b-1 a+b
So if your desired value is b+1 or more, you add 1,2,3, etc lots of the probabilities $\frac{1}{ab}$ until you have the right number of values that at least make your total.
If it's below $b+1$ you need to add $\frac{a}{a+b}$ for each earlier value (moving back through the shaded part until you run into the white parts in the other corner). This sounds complicated because I'm speaking generally, but for any specific instance it's dead simple.
So for your specific case $a=4$ (outcomes in red) and $b=6$ (in blue). You have 24 combinations in your table, each with prob $\frac{1}{24}$, the highest is 10 and you want "7 or more" (more than 6 in your question). This takes you exactly to the "b+1" top right corner:
total combinations
10 1
9 2
8 3
7 4
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... 6 7 8 9 10
For a total of 10 (out of 24) combinations. So the final probability is 10/24.
If it was 8 or more, you'd have 6/24, then 9+ would be 3/24 and so on.
If you had say "5 or more" it would be 10+4+4 combinations or a probability of 18/24 (going back from the corner case of 7 to include exactly 6 and exactly 5). If you go back further still (say "4 or more") you start going down the left arm of the trapezoid and then it's easier to work out the complementary probability ("3 or less") and then subtract from 1.
You can solve quite large problems quickly this way if you know the triangular numbers (1,3,6,10,15,21), but they're also easy to compute if needed.
So say it was d100 + d20, and we wanted a total of 110 or more. That is we want the totals at 110, 111, 112, ... 120, ... that's 11 terms, but they are all after the corner so we want the 11th triangular number ($\frac{11\times 12}{2}=66$) divided by $100\times 20$ so the probability is 66/2000). That's a very rapid calculation!
[It's not much harder even if you go around the corner (below 101); for each one under 101 you just add another 20 combinations; if you get down to small numbers (say below 30 at worst) then just work with the complement.]
