# More dragon egg issues

**URL:** <https://community.gemsofwar.com/t/more-dragon-egg-issues/78755>\
**Category:** Uncategorized\
**Created:** [January 27, 2023, 5:18am UTC](https://community.gemsofwar.com/t/more-dragon-egg-issues/78755 "2023-01-27T05:18:36Z")\
**Posts on this page:** 1\
**Showing post:** 18

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**Author:** ![JamesDurning](https://avatars.discourse-cdn.com/v4/letter/j/e8c25b/32.png) [@JamesDurning](https://community.gemsofwar.com/u/JamesDurning)\
**Post date:** [March 1, 2023, 4:32am UTC](https://community.gemsofwar.com/t/more-dragon-egg-issues/78755/18 "2023-03-01T04:32:50Z")

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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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_[View the full topic](https://community.gemsofwar.com/t/more-dragon-egg-issues/78755)._