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I beleive the Angband RNG is random in the mathematical and statistical sense; but often it seems "unfair" or "streaky" to humans. 
Um. No pseudorandom RNG is "random in the mathematical sense." And there are a whole lot of statistical ways to analyze a sequence for randomness. I suspect the angband RNG is good under some of the basic ones, which is usually good enough.

Well, 1 in 18593 is slightly less probable than flipping a coin and having it come up heads 14 times in a row. I think any intelligent being would look upon that coin with suspicion. That said, I was glad to hear that this RNG had been tested for correlation. Serial correlation analysis is just as important as uniformity analysis when evaluating a RNG and is often neglected.

When evaluating likelihood of an event like repeated fails on a 5% throw, it's better to ignore the first failure, as you are looking at a case where there is at least one failure. So the first failure should be considered a prior. 1 in 18000 isn't all that unlikely when integrated over all games and players.

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I skipped the formulas, of course (maybe Nick can verify them :)). But there are also some human language examplanations and even some code. So I played a bit with it on my machine. If you cast mana storm 100 times, you'll get a streak of 5 or more failures in a row with about 1% probability. 18000 casts give you more than 50% chance of getting a bad run. I guess there is a useful lesson here  always be prepared for the worst :) 
Here is a prettied up version of a python script for calculating this that I found on a link on that site.
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import argparse 
The other thing people often lose track of (or don't think about in the first place) is just how many trials are really being run in a typical game. I'll bet it's a *lot* more than most people realize, so when you do the math and come up with a 1 in 18,000 chance of something, you think "come on, that'll never happen" when in fact there are so many trials that it's to be expected a lot more than you think.

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