Quote:
Originally Posted by whswhs
It's not obvious to me that that's the case. It seems as if the probability of rolling any number is 0, and 0 = 0 = 0.
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In any continuous random variable, the absolute likelihood of rolling any given value is zero. But it's still possible to calculate the probability or rolling higher or lower than any given number.
For example, consider a continuous uniform distribution from 0 to 1. The absolute likelihood of generating value 0.8 is zero, while the probability of rolling less than (or less than or equal to) is 0.8. This can be generalized to p(x)=(x-a)/(b-a), where a is the lower bound of the distribution, b is the upper bound, and x is the upper bound of values we're testing for.
Plug in a=0, b=infinity, and you get p(x)=x/infinity=0. So if you know what the GM rolled, the chance you roll less than whatever the GM rolled is zero. Of course, this is probably wrong for two reasons:
1) Both die rolls are independent events, thus knowing what the GM rolled can have no effect as to whether you roll higher than what the GM rolls.
2) This is coming from arbitrarily plugging in infinity into a formula.
So let's take it from a different angle of attack. Dice of the form 1dX have an average of (X+1)/2, a standard deviation of sqrt((X^2-1)/12)), and the odds 1dX>1dX is (X-1)/(2X). As X approaches infinity, the average and stdev approach infinity, and the odds 1dX>1dX approaches 50%. So save yourself some time searching from your d∞ and toss a coin instead.