It's true that individual actions are very random, but I think Bitty is right about the larger data having average closer to the chance.You can try this out in a spreadsheet. Add a small amount of random numbers between 0-1. The average is probably not very close to 0.5 (the middle point). Now start adding more random numbers and you will see the average getting closer to 0.5 more often.
[p]So I think you can calculate the chance if you have enough data. What is enough data? Depends on the chance, if the chance is smaller you need more data.
How much data you need also depends on how the Random Number Generator works. For instance, taking one regular six-sided die, the RNG most people are familar with, will give you numbers between 1 and 6 with a pretty even distribution that gets closer to 16.66...% as you add more data points.
However, taking two six-sided dice to get random numbers between 2 and 12 will yield a V-shaped curve with the odds of getting a sum of 7 being equal to the combined odd of getting 2,3,11 and 12 combined; of the thirty-six possibilities, six yield a 7 while 2 and 12 only have one possibility each while 3 and 11 each have two, and (1+2+2+1)=6.
So how many data points are needed kind of depends on how many "dice" are used. One gives a linear distribution with equal chances of each result, two results in a V-curve, using three or more is a bell curve. In the event of a curve (V or bell), the center will be halfway between the high end low limits of the range.
Usually rolling a number of times equal to twice the number of possible outcomes will tell you whether it's a single "die" or multiple "dice", but to get a reasonably accurate curve you need a number of samples at least three times the total number of possibilities, an number which increases exponentially as teh number of "dice" increases. For 2d6 with thirty-six possible outcomes, about a hundred data points will be pretty accurate since (3*36)=108, but 3d6 has two-hundred-and-sixteen possible outcomes, meaning over six hundred data points would be required for reasonable accuracy.
The real question then is who (if anyone) has the free time and desire to do the research to make a spreadsheet. I may be a numbers geek, but even I have limits to how much statistics I'm willing to do so it wouldn't be me! Maybe Bittty....
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