This is one of those sleep-deprived middle-of-the-night ideas which I'm reasonably likely to regret posting in the morning once I really wake up - but which, at least at the moment, thinking on my more-corrupted-than-standard hardware, seems like a cool idea.
Most role-playing games have a system for determining whether or not certain actions are successful or not. Most of the time, these can be described as setting a target number, and rolling one or more dice, with various modifiers - eg, you might have to roll a 13 or higher on a twenty-sided dice to correctly answer the sphinx's riddle, and having your handy Book of Ancient Puzzles to refer to may give you a +3 bonus to your die-roll.
How insane and awful an idea would it be to have an RPG system whose core mechanic wasn't based on linear probabilities like that... but, instead, on decibels of Bayesian probability? For example, instead of a bonus adding a straight +3 to a d20, or increasing your odds by 15% no matter how easy the task or how skilled you are, the bonus adds +3 decibels: changing your odds from 50% to 66% if you started out with a middling chance, but only increasing it from 90% to 95% if you're already very skilled.
(And now, back to sleep, and to see how much karma I've lost come the morning...)
One measure of probability is log-odds (or logit), which is the logarithm of the odds-ratio (the base of the logarithm doesn't matter at the moment). That is, for an event with probability p, the log-odds is
And to convert back, for an event with log-odds (base b) of q the probability is
Log-odds are a measure of the amount of information or evidence in support of that event. With a probability of 0.5, the log-odds is 0 (i.e. no net evidence either way), with a probability less than 0.5, log-odds is less than 0 (net evidence against the event), and with p > 0.5, the log-odds is greater than 0 (net evidence for the event).
In the odds world, a bel of evidence means multiplying the odds ratio by 10, so after observing a bel of evidence for it, an event goes from 1:1 to 10:1, or 1:30 to 1:3. In the base-10 log-odds world, this is equivalent to adding 1 to the log-odds, so, those examples become 0 -> 1, and -1.48 -> -0.48. A decibel is adding 0.1 to the log-odds (i.e. a tenth of a bel).
For the example given, 3 decibels turns 0 (log-odds for 50%) into 0.3, and the conversion back gives p = 66%. For 5% the log-odds are -1.279, so 3 decibels turns that into -0.979, which corresponds to p = 9.5%. Similarly, 1% becomes 1.9%, and 0.1% becomes 0.2%.
Thanks!