Let's examine the practical in-game probability "Y" in a typical 4-5 hour DnD game. ("Y" is the number of successful die rolls divided by the total number of die rolls including the misses/failures).
Recall the 95% confidence interval for the in-game probability "Y" (in the normal approximation to binomial distribution):
p - d < Y < p + d
where d = 1.96 * sqrt[p(1-p)/n] and p = theoretical probability.
(For a 99% confidence interval, d = 2.575 * sqrt[p(1-p)/n]).
For this normal approximation to be viable, np(1-p) > 10.
In a typical DnD game, the theoretical probability of success "p" for a d20 roll whether for attacks, skill checks, etc ... is around 50%. (For this normal approximation to binomial distribution to be valid, one needs n > 40 for p=0.5).
If in a typical DnD game a player is doing several dozen or so d20 rolls in the session, let's look at the deviation "d" when n = 41 (ie. borderline case).
So for n = 41 rolls and theoretical probability of success p = 0.5, the deviation "d" of the in-game probability "Y" from "p" for a 95% confidence interval is:
d = 1.96* sqrt[p(1-p)/n] = 0.15
This means that when one is doing 41 rolls of a d20 through a generic 4-5 hour DnD game session, there is a 95% probability that the practical in-game probability "Y" is within the interval 0.35 < Y < 0.65. So 95% of the time, the in-game probability "Y" sees when one is rolling a d20 41 times in a game, can vary from 35% to 65% for a theoretical probability of p=50% (ie. rolling greater than or equal to an 11 for a success on a d20).
If we do this calculation for a 99% confidence interval, the deviation "d" of the in-game probability "Y" from "p" for a 99% confidence interval is:
d = 2.575* sqrt[p(1-p)/n] = 0.20
This means that when one is doing 41 rolls of a d20 through a generic 4-5 hour DnD game, there is a 99% probability that the practical in-game probability "Y" is within the interval 0.30 < Y < 0.70. So 99% of the time, the in-game probability "Y" sees when one is rolling a d20 41 times in a game, can vary from 30% to 70% for a theoretical probability of p=50% (ie. rolling greater than or equal to an 11 for a success on a d20).
Recall the 95% confidence interval for the in-game probability "Y" (in the normal approximation to binomial distribution):
p - d < Y < p + d
where d = 1.96 * sqrt[p(1-p)/n] and p = theoretical probability.
(For a 99% confidence interval, d = 2.575 * sqrt[p(1-p)/n]).
For this normal approximation to be viable, np(1-p) > 10.
In a typical DnD game, the theoretical probability of success "p" for a d20 roll whether for attacks, skill checks, etc ... is around 50%. (For this normal approximation to binomial distribution to be valid, one needs n > 40 for p=0.5).
If in a typical DnD game a player is doing several dozen or so d20 rolls in the session, let's look at the deviation "d" when n = 41 (ie. borderline case).
So for n = 41 rolls and theoretical probability of success p = 0.5, the deviation "d" of the in-game probability "Y" from "p" for a 95% confidence interval is:
d = 1.96* sqrt[p(1-p)/n] = 0.15
This means that when one is doing 41 rolls of a d20 through a generic 4-5 hour DnD game session, there is a 95% probability that the practical in-game probability "Y" is within the interval 0.35 < Y < 0.65. So 95% of the time, the in-game probability "Y" sees when one is rolling a d20 41 times in a game, can vary from 35% to 65% for a theoretical probability of p=50% (ie. rolling greater than or equal to an 11 for a success on a d20).
If we do this calculation for a 99% confidence interval, the deviation "d" of the in-game probability "Y" from "p" for a 99% confidence interval is:
d = 2.575* sqrt[p(1-p)/n] = 0.20
This means that when one is doing 41 rolls of a d20 through a generic 4-5 hour DnD game, there is a 99% probability that the practical in-game probability "Y" is within the interval 0.30 < Y < 0.70. So 99% of the time, the in-game probability "Y" sees when one is rolling a d20 41 times in a game, can vary from 30% to 70% for a theoretical probability of p=50% (ie. rolling greater than or equal to an 11 for a success on a d20).
