More dragon egg issues

18

After pulling yet another dupe today, I realized that I was overcomplicating this problem. If I have X out of 6 dragons after Y pulls, the probability of me getting a new one is (6-X)/6. This is more easily written as a recursive function. Let P(x,y) be the probability of having x dragons after y pulls. Obviously P(1,1) = 1, as a base.
Let’s say I have 4 dragons after 7 pulls. After 8, I have a 2/6 chance of getting 5, 4/6 of remaining at 4.
(x,y) has a (x/6) chance to become (x,y+1), (6-x)/6 chance to become (x+1,y+1). Reversing, we get:

P(x,y) = P(x,y-1) (x/6) + P(x-1,y-1)((6-(x-1))/6)
This is much easier to write as an excel formula.

6 digit probability up to 50 eggs
1 2 3 4 5 6
1 1.000000
2 0.166667 0.833333 0.000000 0.000000 0.000000 0.000000
3 0.027778 0.416667 0.555556 0.000000 0.000000 0.000000
4 0.004630 0.162037 0.555556 0.277778 0.000000 0.000000
5 0.000772 0.057870 0.385802 0.462963 0.092593 0.000000
6 0.000129 0.019933 0.231481 0.501543 0.231481 0.015432
7 0.000021 0.006752 0.129029 0.450103 0.360082 0.054012
8 0.000004 0.002268 0.069016 0.364583 0.450103 0.114026
9 0.000001 0.000759 0.036020 0.277563 0.496614 0.189043
10 0.000000 0.000254 0.018516 0.203052 0.506366 0.271812
11 0.000000 0.000085 0.009427 0.144626 0.489656 0.356206
12 0.000000 0.000028 0.004770 0.101131 0.456255 0.437816
13 0.000000 0.000009 0.002404 0.069806 0.413923 0.513858
14 0.000000 0.000003 0.001208 0.047739 0.368204 0.582845
15 0.000000 0.000001 0.000606 0.032430 0.322750 0.644213
16 0.000000 0.000000 0.000304 0.021923 0.279768 0.698004
17 0.000000 0.000000 0.000152 0.014767 0.240448 0.744632
18 0.000000 0.000000 0.000076 0.009921 0.205296 0.784707
19 0.000000 0.000000 0.000038 0.006652 0.174387 0.818923
20 0.000000 0.000000 0.000019 0.004454 0.147540 0.847988
21 0.000000 0.000000 0.000010 0.002979 0.124434 0.872577
22 0.000000 0.000000 0.000005 0.001991 0.104688 0.893317
23 0.000000 0.000000 0.000002 0.001329 0.087904 0.910765
24 0.000000 0.000000 0.000001 0.000887 0.073696 0.925415
25 0.000000 0.000000 0.000001 0.000592 0.061709 0.937698
26 0.000000 0.000000 0.000000 0.000395 0.051622 0.947983
27 0.000000 0.000000 0.000000 0.000264 0.043150 0.956586
28 0.000000 0.000000 0.000000 0.000176 0.036046 0.963778
29 0.000000 0.000000 0.000000 0.000117 0.030097 0.969786
30 0.000000 0.000000 0.000000 0.000078 0.025120 0.974802
31 0.000000 0.000000 0.000000 0.000052 0.020959 0.978989
32 0.000000 0.000000 0.000000 0.000035 0.017483 0.982482
33 0.000000 0.000000 0.000000 0.000023 0.014581 0.985396
34 0.000000 0.000000 0.000000 0.000015 0.012159 0.987826
35 0.000000 0.000000 0.000000 0.000010 0.010137 0.989852
36 0.000000 0.000000 0.000000 0.000007 0.008451 0.991542
37 0.000000 0.000000 0.000000 0.000005 0.007045 0.992950
38 0.000000 0.000000 0.000000 0.000003 0.005872 0.994125
39 0.000000 0.000000 0.000000 0.000002 0.004895 0.995103
40 0.000000 0.000000 0.000000 0.000001 0.004080 0.995919
41 0.000000 0.000000 0.000000 0.000001 0.003400 0.996599
42 0.000000 0.000000 0.000000 0.000001 0.002834 0.997166
43 0.000000 0.000000 0.000000 0.000000 0.002362 0.997638
44 0.000000 0.000000 0.000000 0.000000 0.001968 0.998032
45 0.000000 0.000000 0.000000 0.000000 0.001640 0.998360
46 0.000000 0.000000 0.000000 0.000000 0.001367 0.998633
47 0.000000 0.000000 0.000000 0.000000 0.001139 0.998861
48 0.000000 0.000000 0.000000 0.000000 0.000949 0.999051
49 0.000000 0.000000 0.000000 0.000000 0.000791 0.999209
50 0.000000 0.000000 0.000000 0.000000 0.000659 0.999341

To answer the OP, after 20 eggs, missing 1 dragon probability is 0.147540, about 1 in 6.778, missing 2 is 0.004454, 1 in 224.5.

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