This was a while ago, but a good way to deal with randomness is to use the 'pick a card' method rather than the 'roll the dice' method.
Thinking of it in terms of a 20 sided die, with the 'pick a card' method you generate a deck with one number for each side of the die, then shuffle the deck. When you need a number, just draw and discard a card. When you run out of cards, shuffle and start over.
With just choosing random numbers, even if the odds are 95% of success, those are only the odds of the average sequence of rolls. You can always get a long sequence of all failures. The fewer rolls there are, the less likely the set of numbers will actually look like the intended average. With coin flips, for example, it wouldn't be surprising to get 10 tails in a row. It's unlikely, (1/1024 odds), but it happens.
With the draw-a-card method, you're guaranteed, over the course of drawing from the deck, to never have large runs of bad (or good) results. With coin flips again, except drawing from a two card deck with one 'heads' card and one 'tails' card, you'll at most get two tails or head results in a row.