What is the average number of draws it takes before you can not draw any more cards from the Deck of Many Things?

Question

When thinking about this question regarding an upper limit on the number of draws, I wondered what would be the expected number of draws you can actually pull off if you called for a large number of draws.

The limiting factors I see are the cards Donjon and The Void which say:

You draw no more cards

...and Talons which would destroy the deck:

Every magic item you wear or carry disintegrates.

The ideal answer would discuss any difference between a 13-card and 22-card deck.

Note after comment by findsul: Assume The Fates card isn't used in connection to your draws.

Answer 1

7.166 for a 22-card deck, 12.5 for a 13-card deck

Xirema has it basically right, albeit approximately. But we can do this without code and get a precise answer.

Take a 20-card deck. Odds of drawing a terminal card are 3/20, so expected number of draws are 20/3, which is 6.66...

Now take a 21-card deck. We have a 4 in 21 chance of pulling either the self-removing card or a terminal, so there are 21/4 (5.25) draws on average until this happens. We then have a 3/4 chance of being done and a 1/4 chance of being in the 20-card deck situation. So expected number of draws to draw a terminal is 5.25 + 1/4(6.66) = 6.9166...

Finally, a 22-card deck gives us 22/5 (4.4) draws until we hit a terminal or self-removing card, so the expected number of possible draws is 4.4 + 2/5(6.9166) = 7.166.

We can do similar for the 13-card example - a 12-card deck averages 12 draws, then the 13-card deck gives you an expectation of 13/2 + 1/2(12) = 6.5 + 6 = 12.5

Answer 2

Technically, you can draw the entire deck if you like.

Others have handled the deep-math version of this answer, and I salute them, but if all you want to do is get as many cards to trigger as possible, the actual upper limit is much simpler

• Declare that you intend to draw the entire deck.
• Wait 1 hour.
• Every card in the deck leaps out and takes effect simultaneously.

It's probably a bad idea.

Your body will be imprisoned in an extradimensional space. Your soul will be imprisoned in an object in a place of the DM's choice, guarded by one or more powerful monsters. An Avatar of Death shows up and immediately becomes frustrated. You receive money, lands, and magic items, all of which immediately turn to dust. Your alignment swaps. You lose 1d4+1 int, and gain +2 to something else. On the bright side, you do get 1d3 wish spells (which cannot save you) and the ability to decide that that was a bad idea and that you shouldn't have done it after all. Perhaps you could use that to exploit the Vizier to get an answer to one question more or less safely? I wouldn't bet on it.

Answer 3

For a 22 Card Deck, the average draw is approximately 7.161 cards

Knowing that the Fools and Jester Cards will remove themselves from the deck if drawn, we therefore know that the odds of drawing one of the "stopper" cards (Talons, Donjon, or The Void) is 3/22, 3/21, or 3/20, depending on whether those two cards have been drawn yet.

So on the first draw, the odds we hit a stopper card is 3/22, or 13.64%. There's a 19/22 chance we did not draw a stopper card, but of those 19 trials, 2 of them now involve a 21 card deck.

So for the second draw, the odds we draw a stopper is (17/19 * 3/22) + (2/19 * 3/21), or about 13.705%; the odds that we therefore draw this stopper after exactly 2 draws is 13.705% * 19/22, which is 11.836%.

We continue this process indefinitely, starting with the third draw where some of the possible decks have 22 cards, some have 21, and some have 20.

For the purpose of giving my calculations a finite stopping point (because hypothetically a person could just keep drawing forever, never hitting one of those cards), I'm presuming that if the odds we haven't drawn one of the stopper cards is less than 1/10000, or 0.01%, then it's okay to simply assume it doesn't happen. So as a result, the margin of error on our mean (which we truncate to 3 decimal digits) is roughly 1 ULP.

Using this process, we can therefore see that the Odds of drawing a stopper card after X draws is this regression:

1 draws: 13.636%
2 draws: 11.836%
3 draws: 10.262%
4 draws: 8.889%
5 draws: 7.692%
6 draws: 6.650%
7 draws: 5.744%
8 draws: 4.958%
9 draws: 4.276%
10 draws: 3.685%
11 draws: 3.173%
12 draws: 2.731%
13 draws: 2.349%
14 draws: 2.019%
15 draws: 1.734%
16 draws: 1.489%
17 draws: 1.278%
18 draws: 1.096%
19 draws: 0.940%
20 draws: 0.806%
21 draws: 0.690%
22 draws: 0.591%
23 draws: 0.506%
24 draws: 0.433%
25 draws: 0.370%
26 draws: 0.317%
27 draws: 0.271%
28 draws: 0.231%
29 draws: 0.198%
30 draws: 0.169%
31 draws: 0.144%
32 draws: 0.123%
33 draws: 0.105%
34 draws: 0.090%
35 draws: 0.077%
36 draws: 0.065%
37 draws: 0.056%
38 draws: 0.048%
39 draws: 0.041%
40 draws: 0.035%
41 draws: 0.029%
42 draws: 0.025%
43 draws: 0.021%
44 draws: 0.018%
45 draws: 0.016%
46 draws: 0.013%
47 draws: 0.011%
48 draws: 0.010%
49 draws: 0.008%
50 draws: 0.007%
51 draws: 0.006%
52 draws: 0.005%
53 draws: 0.004%
54 draws: 0.004%
55 draws: 0.003%
56 draws: 0.003%
57 draws: 0.002%
58 draws: 0.002%
59 draws: 0.002%
>=60 draws: 0.009%
====
Mean: 7.161

^{Code used to produce this output found here: https://godbolt.org/z/t47H5V}

For a 13 Card Deck, the average draw is approximately 12.488 cards

The main differences in a 13-card deck, aside from just being smaller, is the fact that some of the key relevant cards, like Talons and Dunjon (which cause the user to stop drawing cards) or Fool (which is one of the cards that removes itself) are no longer in the deck at all. This has two mechanical changes:

• Drawing a specific card from the deck either has a 1/13 chance or a 1/12 chance, depending on whether Jester has been pulled or not.
• The overall odds of drawing a "Stopper" card has reduced to 1/13 or 1/12, which is much less than the 3/22, 3/21, or 3/20 chances we were dealing with previously.

As a result, this deck tends towards much longer draws by the user.

1 draws: 7.692%
2 draws: 7.150%
3 draws: 6.638%
4 draws: 6.155%
5 draws: 5.702%
6 draws: 5.277%
7 draws: 4.880%
8 draws: 4.510%
9 draws: 4.165%
10 draws: 3.843%
11 draws: 3.545%
12 draws: 3.268%
13 draws: 3.012%
14 draws: 2.774%
15 draws: 2.554%
16 draws: 2.351%
17 draws: 2.163%
18 draws: 1.989%
19 draws: 1.829%
20 draws: 1.682%
21 draws: 1.546%
22 draws: 1.420%
23 draws: 1.305%
24 draws: 1.199%
25 draws: 1.101%
26 draws: 1.011%
27 draws: 0.928%
28 draws: 0.852%
29 draws: 0.782%
30 draws: 0.718%
31 draws: 0.659%
32 draws: 0.605%
33 draws: 0.555%
34 draws: 0.509%
35 draws: 0.467%
36 draws: 0.428%
37 draws: 0.393%
38 draws: 0.361%
39 draws: 0.331%
40 draws: 0.303%
41 draws: 0.278%
42 draws: 0.255%
43 draws: 0.234%
44 draws: 0.215%
45 draws: 0.197%
46 draws: 0.180%
47 draws: 0.165%
48 draws: 0.152%
49 draws: 0.139%
50 draws: 0.128%
51 draws: 0.117%
52 draws: 0.107%
53 draws: 0.098%
54 draws: 0.090%
55 draws: 0.083%
56 draws: 0.076%
57 draws: 0.069%
58 draws: 0.064%
59 draws: 0.058%
60 draws: 0.054%
61 draws: 0.049%
62 draws: 0.045%
63 draws: 0.041%
64 draws: 0.038%
65 draws: 0.035%
66 draws: 0.032%
67 draws: 0.029%
68 draws: 0.027%
69 draws: 0.024%
70 draws: 0.022%
71 draws: 0.021%
72 draws: 0.019%
73 draws: 0.017%
74 draws: 0.016%
75 draws: 0.015%
76 draws: 0.013%
77 draws: 0.012%
78 draws: 0.011%
79 draws: 0.010%
80 draws: 0.009%
81 draws: 0.009%
82 draws: 0.008%
83 draws: 0.007%
84 draws: 0.007%
85 draws: 0.006%
86 draws: 0.006%
87 draws: 0.005%
88 draws: 0.005%
89 draws: 0.004%
90 draws: 0.004%
91 draws: 0.004%
92 draws: 0.003%
93 draws: 0.003%
94 draws: 0.003%
95 draws: 0.003%
96 draws: 0.002%
97 draws: 0.002%
98 draws: 0.002%
99 draws: 0.002%
100 draws: 0.002%
101 draws: 0.002%
102 draws: 0.001%
103 draws: 0.001%
104 draws: 0.001%
105 draws: 0.001%
106 draws: 0.001%
107 draws: 0.001%
>=108 draws: 0.010%
====
Mean: 12.488

^{Code used to produce this output found here: https://godbolt.org/z/npw9w-}