2
\$\begingroup\$

I just "won" at the Deck of Many Things. I'm legit curious. Beyond that actually. What are the odds of what just happened, happening?

(Also, the DM reading my character flaws and letting me find this is one of the lowest moves I've ever seen pulled.)

Context: I just leveled up. I am 0 XP over level 11 (i.e. I have exactly 85,000 XP total). I am True Neutral. I have no money or non-magical items on my person. I can only stop drawing single cards after drawing Talons. Drawing Talons is a Campaign End card unless I've drawn The Moon or The Fates.

Drawing as many cards as possible, what are the odds of drawing a beneficial order of cards from the Deck of Many Things (with a Good End)?

Card class (Good, Neutral, Bad) can change depending on order drawn.

No cards that end in a Bad card class by stopping. I must draw until I hit a stopping condition. Cards are not replaced after being drawn per a homebrew rule, which gives progressively worse odds of drawing a bad card.

To help with the math/reasoning here's a list of cards and card class:

  • The Fates - Good (Negate any single card draw)
  • Vizier - Good
  • Comet - Good
  • Gem - Good
  • Jester - Good
  • Key - Good
  • Knight - Good
  • Moon - Good
  • Star - Good
  • Sun - Good
  • Balance - Neutral
  • Throne - Neutral
  • Ruin - Neutral
  • Fool - Neutral (Bad if drawn after Jester or Sun, draw again)
  • Flames - Neutral (Technically Bad but fine for our case)
  • Rogue - Bad (Unless Moon drawn before stopping)
  • Donjon - Bad
  • Euryale - Bad
  • Idiot - Bad
  • Talons - Bad (Stopping Condition, must have Moon or Fates to become Good End)
  • Skull - Bad (Stopping Condition, Bad End)
  • The Void - Bad (Stopping Condition, Bad End)

Additional Note: Moon gives 3 wishes, which you can use like Fates for our purposes; you cannot wish for more wishes, and you cannot wish it back in the deck. To keep it more simple, we're looking for max benefit with max cards, so using more than a single Wish or Fate to avert Talons (our required stopping condition) is bad.

\$\endgroup\$
  • 1
    \$\begingroup\$ Welcome to rpg.se! Please take the tour and visit the help center when you get a chance. What is this question actually asking? This site is not a forum and doesn't work like one. Are you asking what are the odds of drawing cards in that particular order? If so you will need to provide the order, "beneficial order" isn't a great definition. TBH this feels more like a post to show off then an actual question. Especially since you have done the math already. To save this question remove most of the flavour stuff and reduce it to the actual question you are asking. \$\endgroup\$ – linksassin Jan 18 at 3:01
  • 2
    \$\begingroup\$ @linksassin I'd just add that the OP could certainly pare the post down to an actual question, then present their worked answer to it themselves as an answer post. \$\endgroup\$ – nitsua60 Jan 18 at 3:05
  • 1
    \$\begingroup\$ I was debating on splitting it off. But I thought I recall it being stated somewhere that show proof of work maybe I'm thinking of Overflow. \$\endgroup\$ – Black Jan 18 at 3:07
  • 5
    \$\begingroup\$ Okay, so there's a question hidden in there, I guess. If you're just interested in the probability of drawing the cards in "beneficial order" I think you'd have to define that order for us, and it's not at all clear it's actually an RPG question rather than a math.se question. If there's more to it than that... I don't know? It might be worth popping into Role-playing Games Chat to workshop this a bit before "going live" with it. \$\endgroup\$ – nitsua60 Jan 18 at 3:10
  • 2
    \$\begingroup\$ That would be the beneficial order bit... which linksassin immediately pointed out for removal??!?! \$\endgroup\$ – Black Jan 18 at 3:14
8
\$\begingroup\$

Because there's no card replacement, this actually makes the math a little friendlier than it would otherwise be.

First: Your "Good" runs

The runs that matter require three conditions to be true:

  • They stopped on a "Talons" card
  • Before they stopped, they landed on at least one of either a "Moons" or "Fates" card
  • They did not draw any "Bad" cards at any point in the run

The question is ambiguous as to whether a good run must only contain "Good" cards, or may also contain "Neutral" cards, so I've split the data into two tables: runs that fulfill these requirements that only contain "Good" cards (and "Talons") and runs that contain "Good" or "Neutral" cards (and "Talons"):

\begin{array}{|l|l|l|} \hline \text{# of Good Cards} & \text{Odds} & \text{(as %)} \\ \hline 2 & 1/231 & 0.433\% \\ \hline 3 & 17/4620 & 0.368\% \\ \hline 4 & 16/7315 & 0.219\% \\ \hline 5 & 2/1881 & 0.106\% \\ \hline 6 & 14/31977 & 0.044\% \\ \hline 7 & 13/85272 & 0.015\% \\ \hline 8 & 7/159885 & 0.004\% \\ \hline 9 & 1/101745 & 0.001\% \\ \hline 10 & 1/646646 & 0.0002\% \\ \hline 11 & 1/7759752 & 0.00001\% \\ \hline \end{array}

\begin{array}{|l|l|l|} \hline \text{# of Good/Neutral Cards} & \text{Odds} & \text{(as %)} \\ \hline 2 & 1/231 & 0.433\% \\ \hline 3 & 1/165 & 0.606\% \\ \hline 4 & 183/29260 & 0.625\% \\ \hline 5 & 13/2310 & 0.563\% \\ \hline 6 & 1040/223839 & 0.465\% \\ \hline 7 & 65/18088 & 0.359\% \\ \hline 8 & 611/232560 & 0.263\% \\ \hline 9 & 13/7140 & 0.182\% \\ \hline 10 & 27/22610 & 0.119\% \\ \hline 11 & 5/6783 & 0.074\% \\ \hline 12 & 23/54264 & 0.042\% \\ \hline 13 & 111/497420 & 0.022\% \\ \hline 14 & 13/124355 & 0.010\% \\ \hline 15 & 53/1279080 & 0.004\% \\ \hline 16 & 5/397936 & 0.001\% \\ \hline 17 & 1/447678 & 0.0002\% \\ \hline \end{array}

Whichever kind of run matters, plug the total number of cards into the table, and that'll tell you the odds of drawing a run with that number.

The total odds of drawing only "Good" cards (and "Talons" as a "Good" card) is 1/84 (1.190%), and the total odds of drawing only "Good" or "Neutral" cards (and "Talons" as a "Good" card) is 19/504 (3.770%)

Second: The Raw Odds

I've constructed two tables that relates three things:

  • The number of "good" draws from the run, as you defined
    • The second table counts "Good" or "Neutral" draws
  • The total number of draws from the run, good neutral or bad
  • The odds of any run having that total number of good draws and total draws.

<x,y> in the first column means <good cards, all cards>, and the second column is a fraction representing the odds of drawing that kind of run, along with a decimal number representing its odds.

The rows highlighted in green are the runs that are relevant to your interests, that were used in the above table: the runs where, aside from the "stop condition" card drawn, all of the other cards were "Good". All of the other runs represent the other possible outcomes, where before hitting the stop card, you also hit at least one other bad or neutral card.

This was generated using a C++ program developed to hash out all possible outcomes, running in 40 seconds. The code for this program can be found here

"Good Only" Runs:

Good

"Good or Neutral Only" Runs:

Neutral

\$\endgroup\$
  • \$\begingroup\$ What is that first column? I think you clipped it when editing. \$\endgroup\$ – GreySage Jan 18 at 21:10
  • 1
    \$\begingroup\$ @GreySage Fixed. The post renderer wanted to interpret it as a HTML tag. \$\endgroup\$ – Xirema Jan 18 at 21:13
  • \$\begingroup\$ This is really good so far. More than I was expecting. The only bit left is the conditional nightmare and the fact that the only "Good" stopping condition is Talons with conditions met (if that wasn't clear let me know, I might need to edit again). \$\endgroup\$ – Black Jan 19 at 4:30
  • \$\begingroup\$ @Black Yeah, it might be good to make that clearer in the question. \$\endgroup\$ – Xirema Jan 20 at 1:09
  • \$\begingroup\$ Left the context for Q as-is but changed Q and Talon description to make it clearer. \$\endgroup\$ – Black Jan 20 at 2:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.