Nadir estimation of odds of drawing the lacre

Okay, so this is gonna be a big calculation post. I’m trying to figure out the exact odds of drawing a card in the nadir- specifically, the catafalrique- once.
odds of not drawing the catafalrique with 4-card in the initial hand:
H=15/1614/1513/1412/13 (hand is now full.)
H = 12/16
assuming all cards are 2 irrigo (4 draws before you need to leave):
(12/13)^4= .55
which would mean you’d get at least 1 vial 45% of the time.
Assuming all cards are going to have 1 irrigo choosen except an unlikely garden, the end of battles, and woods in winter, and assuming all choices are success- no second chance failure manipulation, or at least it’s equivalent, so 1.2 average irrigo per discard (so, you have (121+32)/15)=1.2 average irrigo=> 9.6 irrigo in 8 actions 8.4 in 7, halfway between the two is 7.5 draws on average before you need to leave:

opps, should be:
H*(12/13)^7.5 =(41.1% failure rate) → 58.8 % chance of a vial
Assuming you don’t select woods in winter, favoring other cards instead, (131+22)/15 = 1.13333… = ~9/8
H*(12/13)^8 = (~40% chance of failure) → 60% chance of a vial

Odds of not drawing it with a 5-card hand:
Assuming all cards are two irrigo
H*(11/12)^4 = 51.4% chance of a vial
Assuming: see above for 2
H*(11/12)^7.5 =>64.2%
Assuming: see above for 1.1333:
H*(11/12)^8 => 65.7%
Of course, this is assuming you can’t use any second chance discards- i can, but that’d make the math really complex- i think.

Actually, just do the averaging thingy only with the success rate/ second chance tolerance levels. though it will err on the side of overly generous odds.

so, using my reduced persuasive, rounded up to 175 base- the altarful of strangers is .45 (55% failure rate) irrigo, the unlikely garden is .5 irrigo (since after that i’m out of second chances. i’m sure i could manipulate it more effectively by deliberately failing it on successful runs)

with woods:
(22+.45+.5+11)/15 = 1.06 = 8.46 draws:
(12/13)^8.46=> 61.89%
H*(11/12)^8.46=> 67%

without woods
(2+.95+12)/15 = .996666 = 9.03 draws
H*(12/13)^9.03 => 63.5%
H*(11/12)^9.03 => 68.66%

how do we correct this bias? the lazy answer is- we don’t.

I’m feeling lazy, and it’s probably ~2% max. still, these are my estimations, if you notice any errors, please tell me.

TL;DR If you use all three options, and use the second-chance discard mechanic to get rid of unwanted cards, trying to minimize irrigo and removing every card possible from the nadir deck (entring with nightmares less than 2 and a fluke core) you probably have an ~64.2% to 67% chance of drawing the catafalrique with a 5-card lodging, and 58.8% to 61.89% with a 4 card These calculations assume that the second chance discard roughly cancels the initial hand, and that you’re willing to get driven out by irrigo for these results.

if you don’t accept one of the two irrigo cards, the odds improve, up to 65.7% to 68.66%, or 60% to 67% so these would be the odds of a end of battles, or woods in winter normally in a run under these rules.

also, always pack persuasive second chances to force-discard cards, especially at low levels. that is all.
edited by Grenem on 12/16/2015