Normalized Random Numbers (3.5e Variant Rule)

Normalized Random Numbers
The way randomness works means that strings of really good luck and strings of really bad luck are going to happen. People tend to notice the really bad luck more. Human intuition seems to think randomness will play out more like a deck of cards: when a number is rolled it is "used up" and less likely to happen again. This is the gambler's fallacy and a lot of players can get frustrated by it.

The idea behind this rule is simple. Replace your d20s with a deck of cards, removing face cards. When you would roll a d20, instead you draw a card from the deck. Black cards (spades or clubs) are worth +10 (so ace of spades = 11, 10 of clubs = 20, etc.). When the deck is empty, shuffle the cards and start again.

This will normalize luck to some extent. It may be a good idea to restrict it to combat rolls so that the player won't feel like they are "wasting" good rolls on minor roleplaying skill checks.

The rule can be abused with card counting, which is just something you need to consider before adopting it. Card counting can also be a perk, however, since it can add a bit more strategy.

You can use this idea to replace other dice, if you want. Just remove more cards from the deck. For example, a d4 could use the ace to 4 of hearts and diamonds.