Say the player loses a 75% roll, then attempts another identical 75% roll. The game will actually give you better odds for the second roll. I think it's silly -- the game basically lies to you about your odds on the second attempt.
Sid Meier talked about it at length in this video: http://www.youtube.com/watch?v=bY7aRJE-oOY. It starts around 18:25.
> Seeing your five hit attack miss five times at 5% miss
> chance per attack will induce true frustration.
It is. iTunes would play a song, and randomly select the next one (it would do random play, rather than shuffle).
Nowadays it generates a shuffled playlist and goes along, and people whine because it doesn't reshuffle automatically (you have to disable and enable shuffle to reshuffle).
An other option is to use random sampling, and to re-sample after you're done with the current set (possibly excluding any song of the previous sample)
I know there exist "dice" packs of cards, which contain the correct distribution of rolls. If you shuffle a few of those together, you'll probably get a nicer game.
For me, it makes the game feel less "solvable" by basic analysis. Sure, statistically certain observations will hold. For the game you are in, though, you have to be ready to adjust your strategy based on what has happened. (Clearly not in anticipation of future rolls, but more based on the resources you have managed to get, not the ones you wanted.)
Of course, I haven't played at all in a long while.
The solution we settled on was block randomization. But that's probably not what you'd want for a human game as they could potentially figure out the end of each block (and what value remains). But it worked wonderfully for us.
It's why we see stars in the sky, which are more or less distributed randomly in the star field, yet they don't appear random: they appear to form constellations.
It's basically impossible to correct this bad intuition. People educated in the statistics and mathematics of random numbers have to consciously learn what real randomness looks like and override their intuition.
But try as you might, the night sky will still form constellations. And dice rolls will still, frustratingly and confusingly, fall into streaks more often than you'd expect.
Split your class into 2. Give half A the players dice and a piece of paper. Give other half B just a piece of paper. Half B plays with imaginary dice, Half A with real dice. Everyone records 20 games of a piece of paper. You hand in the papers and the demonstrator (or maybe a new one that hasn't been in the class, for extra theatrics) looks at the papers. He will accurately place them into separate piles.
The trick is that fake randomness and real randomness are different enough that they are distinguishable with a high level of accuracy. Real dice will roll the same number in a row far more often than fake dice. The more games you play, the more accurately they can be distinguished.
You can look at the formulas to figure out the odds of rolling the same number 4 times in 100 rolls, but it doesn't drive home the message that your intuition is broken until you see intuition being beaten consistently and reliably by a better tool. It's fun too.
For example, if 20 is rolled then the probability of 20 being rolled again is halved (so it has a 1/40 chance) while the probability of its inverse is doubled (so probability of 1 is 1/10). This dramatically reduces the chance of the three 20s in a row scenario and would very quickly regress to the mean, just like people expect. Added flair for affecting the probabilties of all possible values subtly (so that after rolling a 20, a 2 becomes more likely and a 19 less likely).
I'm sure I can't be the only person to think of this.