Chances of winning unimproved with 99

So I am talking to someone on the forums who likes to tell me I am smoking crack and that raising 99 from EP is never correct. So of course, I throw the math back in his face.

and since your chances of winning with 99 unimproved are miniscule, why not raise hands like 88 or 77 too, by that logic?

Okay. Let's do some math. Let's look at 99. There are 28 unseen cards between 2 to 8. The probability of flopping an overpair is 28/50 * 27/49 * 26/48 = 0.167. So there is a 16.7% chance of flopping an overpair, and a roughly 11% chance of flopping a set.

Now let's look at 88. There are 24 unseen cards from 2-7. So our chances of making an over pair is 24/50 * 23/49 * 22/48 = 0.103 or 10.3%, with the same 11% chance of hitting a set. So the winning chances between 99 and 88 drop dramatically. 77 drops to a 5.8% chance of making an overpair, and so 77 has extremely low chances of winning unimproved, where 88 has decent but still low chances. 99 seems to be a cutoff, and against certain players and situations becomes profitable, even raising.

So winning unimproved for 99 is not miniscule. It's low compared to TT, JJ and up, but it happens enough times that raising can become profitable.