This is an important observation for game design and game play. We've all seen the player who has rolled several low scores on to hit rolls in a D&D session who says "the odds are getting better of me rolling a 20" or the player who has rolled 6 "aces" in a row in Savage Worlds who picks up the dice and says "the odds of me acing again are 1/(some huge number)." In both cases, the individual is wrong. While it is true that given a sufficiently large draw that the die rolls of a player will tend toward the mean, prior die rolls have no influence on future die rolls. As an extension of that, the player who has already rolled 6 "aces" has exactly a 1/6 chance (assuming a d6 is being rolled) of acing on the 7th roll. The prior rolls have no influence over the initial roll. The answer would be different if the person had stated before rolling at all that there chances of acing 7 times was 1/(some huge number) but it isn't true after the person has successfully aced 6 rolls and is now rolling the seventh roll.
When I was a 21/Craps dealer as an undergrad in Nevada, I saw how this kind of flawed logic could have real financial consequences.
"Wow!" The player would say, "there have been a lot of 7's rolled in a row, so it's time to 'buy' the 4 at a 5% vig." Their underlying assumption is that prior rolls affect future outcomes in die rolls. They don't.
Interestingly, when players are in situations where prior decisions DO affect future outcomes they are just as prone to intuitively come to the wrong conclusion. A great example of this phenomenon is the Monty Hall problem where a player is given three choices, shown the results of one of the selections they did not make, and then asked to either switch or keep their original choice. The correct answer to this question - because prior choices DO affect outcomes in this case - is counter intuitive. I'll let the good folks at Khan Academy explain why.
Think about how this dilemma will affect game play in hidden information games that you design and play. And let this be a reminder that understanding how a probabilities work can make you a better player, designer, or game master.