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It's pretty common knowledge that different Pokémon have different base call rates, that lowering their HP increases the call rate, and that inflicting status lowers the call rate to 0, and I've also heard from a few sources that using supereffective attacks increases the rate as well, but I've never seen an actual formula.

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The best choice of action is to look on Serebii. Serebii has information on what routes have which Pokemon, and how often they appear based on percentage. They also have a a rate of which Pokemon comes to help from an SOS battle and the percentage of which Pokemon is called from that encounter too.
Moved this to a comment. The question asks how likely it is the Pokémon will call for help in the first place, not for the chances specific Pokémon will appear.
I believe there isn't a formula but if you want a list:
https://pastebin.com/Uv3k3QYW

It is a regular site.
That's a great start, but what do those numbers mean? How does the game use those numbers? And how much more likely is it if I lower their HP? If I use a supereffective move, how much more likely does that make its call rate, and for how long? Just the next turn? Five turns? The rest of the battle? These are the kinds of questions I'm looking to answer, and I find it very strange nobody else has figured it out or datamined it yet. At least, not as for as I've looked.

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Each Pokemon has a base call rate of either 0, 3, 6, 9, or 15. 0 is reserved for extraneous species that get called in, like Mareanie or Blissey. Of course, there's also no chance of a call if two opponents are still on the field, if the only opponent is inflicted with a status, or (in USUM only) if there's already been one successful call in this chain and you haven't used an Adrenaline Orb yet. Anyway, the "call rate" represents the percentage of time it will call for help provided it's able to do so, at the end of a turn where it's in green health and you have not yet used an Adrenaline Orb. That percentage can only have two possible types of multipliers:

  • green health, with Adrenaline Orb: x2
  • yellow health, no orb: x3
  • red health, no orb: x5
  • yellow health, with orb: x6
  • red health, with orb: x10

As such, a 15-call-rate Pokemon is guaranteed to call for help every single turn in the 10x case, unless it's prevented from doing so by having a status or having the second slot survive.

Separately from that, there's the chance that the call actually succeeds. By default, the 3/6/9/15 call rate groups have success rates of 12/24/36/60, and being in yellow/red heath doesn't improve those chances at all. The oft-cited Pressure, Intimidate, and Unnerve abilities don't actually affect the call rate, but they do affect the odds of success, giving it a 20% boost to 14.4/28.8/43.2/72% respectively. The only other bonus that can improve this chance is the "momentum bonus":

  • opponent called for help last turn but "Its help didn't appear!": x4.5
  • opponent called for help last turn, and it was answered, but you promptly OHKO'd it with a super-effective hit: x3
  • opponent called for help last turn, and it was answered, but you KO'd it with something that wasn't super effective: x1.5

Using an item, switching up which side of the SOS battle is the "caller" (if you're going for a non-Harvest approach), or just biding time while you wait for the new opponent to do something that might identify what ability they have, all forcefully bring your momentum bonus to a close, and you have to go without that bonus until you get another successful call. 15-rates with the red health, orb, and Pressure bonuses just need 1.5x momentum to make sure that not only do they call for help every turn without fail, but those calls are also answered without fail, so you don't need to worry about lining up type matchups there. But for lower call rates, the x3 is a big benefit to helping the chain go faster, if you have a suitable SE move to mow everything down in rapid succession.

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Curious how you found this info? It seems to match my own experiences with SOS chaining, but I have never heard of mechanics like the momentum bonus.
Hugely useful if this is accurate.
Thanks for reading my stuff on Smogon and answering.
By any chance is anything you said in this answer only true for SM or only true for USM?
The only changes in SM are that wilds aren't capped at a single call if you haven't used an Adrenaline Orb yet, some Pokemon such as Chansey had their call rates increased from or decreased to 0, and they only devoted 1 byte instead of 4 to the SOS chain counter, which results in the shiny/HA/minimum IV level bonuses resetting after every 256 successful calls.
I was asking this because Bulbapedia lists different multipliers for SOS call success rates. Are you sure you're right and not Bulbapedia?