Actions which give out Eyeless Skull.
For a list of uses (cards/storylets it unlocks, and actions requiring it), see Category:Eyeless Skull.
Starting the expedition usually gives 1 progress; you need to raise 19 more progress (to a total of 20) to complete it. More rarely, 17.43% of the time, the starting action gives 3 progress, and you need to raise 17 more. With Watchful 267+, A buccaneering approach gives 3 progress/3 supplies/action. When the Airs values are right (96-100), the more efficient A sign? gives 4 progress/action. The fastest (though not the cheapest) method of obtaining supplies gives 2.5 supplies/action.
On average, then, one expedition takes 1 action to start, (17*17.43%+19*(100-17.43)%)/(3*95%+4*5%)=6.12 actions to progress and 1 action to complete. It also takes (17*17.43%+19*(100-17.43)%)-(4*5%)=18.45 supplies, which must be obtained over at least 18.45/2.5=7.38 actions. This gives a total of 13.5 actions per expedition.
Around 9.92% of these expeditions result in a skull.
On average, then, finding one shrine with a skull takes 136.1 actions (for Watchful 267+, and longer for lower).
See the sights, at Airs 61-70, gives one skull on 3.22% of its successes. Wander the Quarter, at Airs 21-40, gives one skull on 2.73% of its successes. See the sights also gives +1 CP Nightmares on every success. If you cycle Airs using the other options on Seeking Curios (and avoid looking for evidence of the Correspondence), you will also gain +1 CP Nightmares from 50% of poking about among the ruins at Airs 0-20, and +1 CP Nightmares from the 4.99% success on Explore the darker corners at Airs 41-60. Nightmares can be cured at 10 CP/3 actions by using a 1 chess + 2 confessions combo with a friend.
So, with Watchful 250+, the chance of hitting the right Airs and then obtaining a skull is 10%*3.22%+20%*2.73% = 0.868%, making for one skull every 115.2 actions. Of these 115.2 actions, 10% will be See the sights, 20% will be Poke about among the ruins, and 20%*4.99% will be rare successes from Explore the darker corners, making for at least 115.2*21%=24.19 CP Nightmares per skull, which can be cleared in 24.19*3/10 = 7.26 actions.
On average, then, finding one skull in the Quarter at large takes 122.5 actions (at Watchful 250+, and longer for lower).
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