Now that you’ve woken up, why not try showing the math? E.g., why is a change from 50% to 66% “3 decibels” ? What would happen if you added 3 decibels to a previously “impossible” task (5%? 1%? 0.1%?)
Would this offer any benefit in terms of either fun or realism beyond the fudge dice, 3d6, etc. from other people’s comments? What benefit? If the benefit is educational, what exactly would people learn?
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 frac{b{q}}{1b{q}}
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%.
Now that you’ve woken up, why not try showing the math? E.g., why is a change from 50% to 66% “3 decibels” ? What would happen if you added 3 decibels to a previously “impossible” task (5%? 1%? 0.1%?)
Would this offer any benefit in terms of either fun or realism beyond the fudge dice, 3d6, etc. from other people’s comments? What benefit? If the benefit is educational, what exactly would people learn?
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 frac{b{q}}{1 b{q}}
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!
Here you go.