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The 0.02% chance per card mentioned above is wrong. IIRC, it's 0.3% per card from an arcana pack to be a Soul card. I don't remember if spectral cards from packs also have a 0.3%. For some reason I thought it was better than 0.3%. Don't know. It's at least 0.3%, for sure.
Also, gold stake now has a mechanic that, if you find a Soul card, it'll spawn a legendary joker that you DON'T have a gold sticker on in your collection yet, assuming you don't have all 5 yet. So if you have access to gold stake, you might consider hunting for a Soul card on that stake.
If you don't play gold stake, that 5th legendary could take a while. I found my 5th on my 18th Soul card, which was not lucky but could have easily been worse. I think I calculated it at one point and about 5% of players will need more than 18 Soul cards.
Good luck!
Only way to improve the odds beyond that is by modding the game, but that's cheating. Up to you OP to decide if you want that, or want to spend a potentially long time trying to get that last card. You could just grind your way up to gold on the various decks in the meantime if you're interested in Completionist+.
Supposing you get 4 cards per arcana pack on average, the odds of ZERO Soul card in 5,000 packs are
less than 1 in 1e26
I could add a comment, but it would be against Steam ToS.
Some extremely unlucky outliers will eventually happen as long as people keep playing. Of course this would be a ridiculously rare happening, but some graphs I've seen in poker made me a believer.
Less likely than winning a 1 in 100M lottery, 3 tickets back to back.
Or if you like poker, shuffle a deck, draw 5 cards and it's a straight flush. FIVE TIMES IN A ROW. The odds of that are better than 1 in 1e26.
EDIT: and I'm sure someone can open 5k arcana packs in 110 hours, right? We can expect what, maybe 2 such packs per full ante level? So that's 2.5k ante levels, IF YOU BUY ALL PACKS, which never happens because of early ante and limited money. Or the equivalent of 312 full, completed 8 ante runs. In 110 hours. Someone's a top tier speedrun competitor!