Major Multipliers ... 2s wild vs standard

[Numbers Guy]

My VP-playing friend and I are having a friendly argument over which version of Major Multipliers produces the multipliers more often. Not which game has a higher RTP, but simply which one generates more multipliers.

In the non-wild-card games, you get the multipliers if you are dealt jacks or better, two pair, or three of a kind. I believe I have the rough math correct on this one:

You are dealt a specific pair (and only that pair) about 3.99% of the time. So you should get Jacks/Queens/Kings/Aces about 4 x 3.99% or 15.96% of the time.
You should be dealt two pair about 4.75% of the time.
You should be dealt trips about 2.11% of the time.

Adding those three together, you should generate the multipliers about 22.82% of the time.

In deuces wild, you get the multipliers only if you are dealt three of a kind and nothing else. Therefore, you have to add the times you get natural trips (other than deuces), the times you get a pair and a deuce, and the times you get two deuces and nothing to go with them.

The natural trips is the easy part, that's about 1.95%. It's the rest that I'm struggling with a little bit.

All told, I would like to know if the sum of those deuces wild situations is greater than 22.82%. Intuitively, I don't think it is, but my friend seems confident that it is, so if anyone can settle this friendly argument, I'd be very grateful!