I think that the discussion of TrynnaDoStat's answer illustrates the point well: we use simulations whenever the problem is impossible to solve analytically (i.e. the posterior distributions of parameters in a hierarchical model), or when we're simply too annoyed to put the time into working out the solution analytically.

Based on what I've observed on this website, the threshold of "annoying enough to simulate" varies widely between statistician. People like @whuber can, apparently, glance at a problem an immediately see the solution while mere mortals like myself will have to carefully consider the problem and maybe do some reading before writing a simulation routine to do the hard work.

Keep in mind that simulations aren't necessarily a panacea since with large data sets, or complicated models or both you'll spend enormous amounts of (computer) time estimating and checking your simulation. It's certainly not worth the effort if you could accomplish the same goal with an hour of careful consideration.

I think that the discussion of TrynnaDoStat's answer illustrates the point well: we use simulations whenever the problem is impossible to solve analytically (e.g. the posterior distributions of parameters in a hierarchical model), or when we're simply too annoyed to put the time into working out the solution analytically.

Based on what I've observed on this website, the threshold of "annoying enough to simulate" varies widely between statistician. People like @whuber can, apparently, glance at a problem and immediately see the solution while mere mortals like myself will have to carefully consider the problem and maybe do some reading before writing a simulation routine to do the hard work.

Keep in mind that simulations aren't necessarily a panacea since with large data sets, or complicated models, or both you'll spend enormous amounts of (computer) time estimating and checking your simulation. It's certainly not worth the effort if you could accomplish the same goal with an hour of careful consideration.