Upgrade your account to a Community Supporter account and remove most of the site ads.

Reply to thread

Message: <blockquote data-quote="tarchon" data-source="post: 1820813" data-attributes="member: 5990">There isn't really a single method for all cases, though there are a number of theorems that can be useful.Generally, when you have a simple sum of independent random numbers (like a sum of die rolls), there's a very useful theorem which says that the probability mass function (PMF) of the sum is the convolution of the PMFs of the addends.  This sounds complicated perhaps, but it's very simple to apply with dice, since they have uniform discrete distributions.  You can do it manually, but it's well suited to computer code, and there are extremely efficient algorithms for it.  Also, if you happen to know the characteristic function (CF) of the addend distributions, the CF of the total is just the CFs of the addends multiplied together.Once you have the PMF of the distribution, it directly gives you the answer to "how likely is n?"Things only get complicated if you do things like dropping the lowest die or rejecting characters with a score less than 5, or something like that.  In those cases, you can sometimes come up with elegant expressions, but it often takes a lot of work and the expression is usually something that's prohibitively difficult to calculate without a computer.  Therefore, unless you have some overwhelming need to have the formula written down, some combination of some elementary theorems like the one above and brute force is usually the easiest practical way to get there.   Often even simple enumeration and testing of the cases takes a trivial amount of computer resources, and even when it takes too much memory or time to do that way, some very simple arguments can be used to reduce it to manageable levels.In my experience with dice probabilities, I almost always recommend something very close to simple enumeration because, invariably, once you have the answer to one form of the problem, somebody changes it slightly, and it's a lot easier to change enumeration code that runs in 10 seconds than to spend 20 hours coming up with a new formula that takes a computer to run anyway.</blockquote>

Verification