I Odds of getting pairs -- QM version

[anuttarasammyak]

As a spin out of How to calculate these odds?
https://www.physicsforums.com/threads/how-to-calculate-these-odds.1012014/
I have a question for quantum objects.
Question : An atom system has 16 different energy levels filled with 16 electron pairs of opposite spin direction. We pick up or measuring energy of 4 of them. Calculate odds for numbers of pairs of same energy we get in the observation results.

I assume:
If we observe spin z component of electrons together, the result is same as the thread.
If we do not observe it the number of cases are
2 pairs : $\ _{16}C_2 = 120$
1 pair : $\ _{16}C_1 \ _{15}C_2 = 1680$
0 pair : $\ _{16}C_4 = 1820$
Total case number is $120+1680+1820=3620$
So the probabilities are
P(2)=0.033 > 0.003 which is the result in the thread
P(1)=0.464 > 0.187
P(0)=0.503 < 0.810

I would like to know whether such a difference between classical and quantum cases takes place or not and if yes also this calculation is reasonable or not.

 

[bland]

Ha, I like it. subbed.