A few days ago I came across this question and thought it might be interesting for some to shed a bit more light on this. The answer is actually quite complex and I want to break it down into the following bits:

- Deck composition
- Frequencies and probabilities
- Expected wait times

Deck composition

Establishing the exact composition of your deck is easy in theory but quite laborious in practice. It’s easy because “all” you have to do, is establish which qualities you have and then compile a list of cards that match your qualities. But the sheer number of qualities and cards in the game make it very time consuming without some sort of automation. The best I could do is use my own deck as a semi-typical example. I’ll include the list in the spoiler and you can use it as a sort of a template. Beware, though - it takes a long time!

The first list is all the cards that I either can’t remove or require a lot of fate to remove or are very useful cards that you could in principle remove but the resources required to get them back later would render any savings moot.

- Cards that virtually all sufficiently advanced players will have

40 Standard

A day at the races

An unusual wager

Give a Gift! A commotion in the Square of Lofty Words

A Sporting Sort

Mr Wines is holding a sale!

Wanted: Reminders of Brighter Days

Investigating the Affluent Photographer

All fear the Overgoat!

Reeducating Lyme

The Phantom of the Antimacassar

Revisit the Theosophisticals

The Geology of Winewound

Investigate Doctor Schlomo

Visit

The mechanics of progress

One’s public

The Northbound Parliamentarian

Back to the Palace cellars

Weather at last

The Awful Temptation of Money

A past benefactor

A Polite Invitation

Riding your Velocipede

The Soft-Hearted Widow

4 Mysteries of the Neath cards (Devices and desires etc)

12 Faction cards

1 Infrequent

A visit from Slowcake’s Amanuensis

7 Very Infrequent

An unsigned message

A merry sort of crime

A Presumptuous Little Opportunity

4 Relicker cards

4 Unusual

Below the Neath

The Paronomastic Newshound

A dream about a window at night (assumed Unusual)

A dream about the mist (assumed Unusual)

4 Rare

A disgraceful spectacle

A voice from a well

2 Moods cards

- Optional POSI Gear cards

9 Standard

More Larks with the Young Stags

Out and About on your Ratwork Velocipede

Once Upon a Time in a Carriage

A day out in your Clay Sedan Chair

A library of your own

Your Edifice of Black Stone

A Day with God’s Editors

The Life of Crime

A Night with your Glabrous Companion

1 Very Infrequent

Swap Incendiary Gossip

- Lodging cards

8 Standard

The Listing Tower: Intrigue in a Half-Abandoned Mansion

The Sleepless Tower: Disturbances at a Cottage by the Observatory

The Heron Tower: Events at a Lair in the Marshes

The Tower of Sleeping Giants: The Secrets of the Rooms above a Bookshop

The Tower of Sparrows: Vice and Virtue in a Gambling Den

The High Castle: What Occurs in a Rooftop Shack

The Windward Tower: the Matter of the Decommissioned Steamer

The Tower of Knives: Difficulties at a Smoky Flophouse

4 Frequent

The Tower of Eyes: Behind Closed Doors at a Handsome Townhouse

The Western Tower: a Guest Room at the Brass Embassy

The Tower of Sun and Moon: a Reservation at the Royal Bethlehem Hotel

The Lofty Tower: the Potential of Premises at the Bazaar

- Story “bookend” cards

6 Standard

I’ve brought something for you to try, dear (Aunt)

Rat Melancholy

Restoring souls

The Mayor of London

A Tale of the Generous Princess

Invited to another revel of the Masters

The second list includes cards that I can remove from my deck relatively easily

- Menace cards (under 6)

4 Standard

An afternoon of good deeds?

The Law’s Long Arm

The vigilant gentlemen in blue

A Restorative

1 Ubiquitous

A merry gentleman

1 Very Infrequent

A Moment’s Peace

1 Unusual

The Interpreter of Dreams

- City Vices

5 Standard

City Vices: a Rather Decadent Evening

City Vices: an Entanglement with an Old Friend

City Vices: ask the Sardonic Music-Hall Singer to help you

City Vices: help the Sardonic Music-Hall Singer

City Vices: Orthographic Infection

1 Unusual

City Vices: a tournament of weasels!

- Other removable cards

4 Standard

Connected pet

An Implausible Penance

A consideration for services rendered

A parliament of bats

1 Infrequent

A libraryette for Mr Pages

1 Unusual

Pass the Cat: a wriggling delivery

Finally, there are the 12 Conflict cards. 5 are Standard and 7 are Very Infrequent, although once the conversions are complete it’s likely all will be Very Infrequent. I typically have 1 Standard and 2 VI conflict cards “live” in my deck.

Frequencies and probabilities

Each card in Fallen London has a “Frequency”. Each Frequency is associated with a certain number of copies of a unique card that is added to the deck. For example, in my deck above, each Standard card will actually have 100 copies, while each Rare card will have 10 copies. The details for each frequency can be found here.

Since the overall composition of the deck changes constantly, the actual probability of drawing a given card fluctuates even though its nominal Frequency stays constant. It is time consuming but nevertheless possible to calculate the actual probabilities provided you have the exact deck composition. Again, in a general case it’s hard without automation but I will use my deck above and walk through some examples.

Example 1 will be the deck consisting of all the “guaranteed” cards (the first list in the spoiler above). This is the most trimmed version of my deck. It is only achievable with a Remote Address and so implies a hand size of 3. Example 2 will be the same deck but with non-Menace City Vice cards added in. I will assume that I am now free to have a had size of 5. Example 3 will be the same as Example 2 but now also all the cards from the second list as well as 3 conflict cards (1 standard and 2 very infrequent).

Step 1 - establish complete breakdown of unique cards in the deck by frequency.

Note that the hand size affects the deck composition as you can park some cards (which I’ll assume are Standard) in your hand. So simply adding up all the relevant categories in the list above, I obtain the following breakdown for Example 1:

Rare - 4 unique cards

Unusual - 4 unique cards

Very Infrequent - 8 unique cards

Infrequent - 1 unique cards

Standard - 63 - 4 = 59 unique cards (4 cards in hand)

Frequent - 4 unique cards

Step 2 - multiple unique counts by the relevant frequency and add these all up to give the total number of cards in the deck.

For Example 1 we have:

Rare - 4*10=40
Unusual - 4*20=80

Very Infrequent - 8

*50=400*

Infrequent - 180=80

Infrequent - 1

Standard - 59

*100=5900*

Frequent - 4200=800

Frequent - 4

Total: 7300 cards in the deck

On the other hand, Example 2 will have 7920 cards, while Example 3 will have 10190 cards.

Step 3 - calculate the probabilities (in percentage terms). To do this multiple the Frequency by 100 and divide by the total number of cards.

So in Example 1, the chance of drawing a particular Standard card is 100*100/7300=1.37%, while the chance of drawing a Rare card is correspondingly 0.137%. If we now add the City Vice cards (Example 2) then the chance of drawing a Standard card goes down to 100*100/7920=1.26% and the chance of a Rare card goes down to 0.126%. Finally if we expand our deck even more by adding menaces, conflicts etc (Example 3), Standard rate goes down to 100*100/10190=0.98% and Rare rate goes down to 0.098%.

Expected wait times

The examples above should show that the deck composition can change the probabilities significantly. However, the human brain isn’t very good at reading probabilities and a different metric might be more helpful.

The reason we care about card probabilities is because we want to know when we are going to draw a particular card we are waiting for. So a useful metric is “Expected wait time”, i.e. how many draws on average do we have to make before the card we are waiting for comes up. Once we have the probabilities, this is really easy to calculate - simply divide 100 by that probability.

So in Example 1, the expected wait time for a standard card is 100/1.37=73. So you’d expect to have to draw 73 cards before the card you are waiting for comes up. In other words, for a dedicated player who spends all of their actions, they should expect to draw such a card twice a day. The expected wait time for a Rare card is 730. So a Rare card should appear once in 5 days (in this maximally trimmed version of the deck).

By contract, in Example 3, the expected wait times for Standard and Rare cards are 102 and 1020. You’d draw a standard card once or so each day but the Rare card would take as much as a full week. So you lose 2 days, on average, when fishing for a Rare card without trimming your deck.

Now, you may say that once a week for a rare card in a bloated deck is overly optimistic. There are three things to bear in mind:

- A deck may be even more bloated than mine in Example 3
- The expected wait time is just an average. When dealing with randomness you also have to deal with the concept of “variance”. I won’t get into this here but suffice to say that if a Rare draw has probability 0.098 and expected wait time of 1020 then you should not be surprised if you actually have to wait as much as 1020*(1+2*sqrt(100-0.098)/10)=3059 or in other words 3 weeks.
- Even unexpected results can happen. If you find yourself waiting for 2 months for a rare card that’s not, statistically speaking, all that implausible. Some other time you may find yourself drawing 3 Moods in a single day!

Hopefully some of you would find this interesting or insightful.

ETA: corrected calculation

edited by genesis on 12/18/2016