Note the symmetry factor with the factorials: we’re computing the probability of the observed counts, not the probability of a particular string of outcomes, so we have to add up probabilities of all the outcomes with the same counts.
Can you clarify why we look at the probability of counts rather than the particular string?
The reason I’m asking is that if a problem has continuous outcomes instead of discrete then we automatically look at the string of outcomes instead of the count (unless we bin the results). Is this just a fundamental difference between continuous and discrete outcomes?
It’s not really important, all that matters is that we’re consistent in which one we use. We have to always include the symmetry factor, or never include it.
In this case, I went with counts because our data does, in fact, consist of counts. Because we’re assuming each die roll is independent, we’d get the same answer if we just made up a string of outcomes with the same counts, and used that as the data instead.
Can you clarify why we look at the probability of counts rather than the particular string?
The reason I’m asking is that if a problem has continuous outcomes instead of discrete then we automatically look at the string of outcomes instead of the count (unless we bin the results). Is this just a fundamental difference between continuous and discrete outcomes?
It’s not really important, all that matters is that we’re consistent in which one we use. We have to always include the symmetry factor, or never include it.
In this case, I went with counts because our data does, in fact, consist of counts. Because we’re assuming each die roll is independent, we’d get the same answer if we just made up a string of outcomes with the same counts, and used that as the data instead.