My question is about when not to use Bose-Einsten statistics (otherwise known as stars and bars). Of course it shouldn't be used when order matters or you can not sample with replacement. However, while reading Blitzstein's textbook it is noted:
"The Bose-Einstein result should not be used in the naive definition of probability except in very special circumstances"
where the $P(A)$ under the naive definition is defined to be $\frac{|A|}{|S|}$.
It then proceeds to give an example:
"As another example, with $n = 365$ days in a year and $k$ people, how many possible unordered birthday lists are there? For example, for $k > = 3$, we want to count lists like (May 1, March 31, April 11), where all permutations are considered equivalent. We can't do a simple adjustment for overcounting such as $n^k /3!$ since, e.g., there are 6 permutations of (May 1, March 31, April 11) but only 3 permutations of (March 31, March 31, April 11). By Bose-Einstein, the number of lists is ${n + k -1}\choose k$. But the ordered birthday lists are equally likely, not the unordered lists, so the Bose-Einstein value should not be used in calculating birthday probabilities."
This doesn't make sense to me. Of course, the ordered birthdays are equally likely. Why wouldn't the unordered birthdays also be equally likely? Could someone explain this to me and potentially provide another example for me to see the limitations of Bose-Einstein? Thank you in advance!
