# Hey Buddy! Doing the Dampe Math

If you’ve ever watched an Ocarina of Time 100% speedrun, or played the game yourself, you’ll probably know about Dampe. As a gravedigger in Kakariko graveyard, he’ll offer to dig up graves for you when you go to him as child Link to get a random treasure. If you’re mildly lucky, you’ll get enough rupees to try again. If you’re really lucky, you’ll get the heart piece.
The latter is obviously part of the 100% definition of the game, so hundo speedruns sooner or later have to face dampe and gamble on the 10% chance of him digging up the heartpiece.

As one of the few major rng-dependent1 events in the run he’s kind of notorius, especially when he kills multiple runs in a row by needing more than 10 attempts to finally get the heart piece.
The math itself is pretty straight forward: In a speedrun, runners use a method to reset Dampe to a specific plot, allowing them to dig on the same grave spot over and over again, theoretically ad infinitum. While this method is not as “safe” as the intended method, which is capped at 15 tries maximum, it’s significantly faster unless you get “bad Dampe rng”. Each time Dampe digs, there’s a 10% chance to get the heartpiece, which means in mathy terms we can define $$p = 0.1$$ and $$k$$ as the number of tries until success, giving us:

$$P(k = 1) = p = 0.1$$

Second try Dampe would be the probability of “failing” the first attempt and then succeed afterwards, or:

$$P(k = 2) = (1-p) \times p = 0.9 \times 0.1 = 0.09$$

And in general terms:

$$P(k) = (1-p)^{k-1} \times p$$

Which, conveniently, is known as the geometric distribution. In R, we can calculate individual probabilities using dgeom(x, prob = .1) to get the probability for “failing x times before success”, meaning we need to use k = x - 1 to get the number of attempts it took to get the heartpiece, include the last dig.

So, with a lot of talk about odds and chances and whatnot every time Dampe is misbehaving again, I thought I’d take the opportunity to make a little lookup table to see just how unlikely your latest 61st try Dampe really was.

Here’s what’s included:

• Attempt: How often you had to let Dampe dig until the heart piece, $$k$$
• Probability: The probability for $$k$$, as a singular outcome
• Odds (Singular): $$\frac{1}{\text{Probability}}$$, the probability for that outcome in human readable terms
• Probability (cumulative): Probability of $$1$$ through $$k$$ tries, i.e. the probability for getting $$k$$ or lower tries

(So, in statsy terms, I’m basically just printing the pmf and cdf for $$k = 1$$ through $$k = 105$$ in a table. Yip.)

Attempt Probability Odds Probability (cumulative)
1 0.1000000 1 in 10 0.1000000
2 0.0900000 1 in 12 0.1900000
3 0.0810000 1 in 13 0.2710000
4 0.0729000 1 in 14 0.3439000
5 0.0656100 1 in 16 0.4095100
6 0.0590490 1 in 17 0.4685590
7 0.0531441 1 in 19 0.5217031
8 0.0478297 1 in 21 0.5695328
9 0.0430467 1 in 24 0.6125795
10 0.0387420 1 in 26 0.6513216
11 0.0348678 1 in 29 0.6861894
12 0.0313811 1 in 32 0.7175705
13 0.0282430 1 in 36 0.7458134
14 0.0254187 1 in 40 0.7712321
15 0.0228768 1 in 44 0.7941089
16 0.0205891 1 in 49 0.8146980
17 0.0185302 1 in 54 0.8332282
18 0.0166772 1 in 60 0.8499054
19 0.0150095 1 in 67 0.8649148
20 0.0135085 1 in 75 0.8784233
21 0.0121577 1 in 83 0.8905810
22 0.0109419 1 in 92 0.9015229
23 0.0098477 1 in 102 0.9113706
24 0.0088629 1 in 113 0.9202336
25 0.0079766 1 in 126 0.9282102
26 0.0071790 1 in 140 0.9353892
27 0.0064611 1 in 155 0.9418503
28 0.0058150 1 in 172 0.9476652
29 0.0052335 1 in 192 0.9528987
30 0.0047101 1 in 213 0.9576088
31 0.0042391 1 in 236 0.9618480
32 0.0038152 1 in 263 0.9656632
33 0.0034337 1 in 292 0.9690968
34 0.0030903 1 in 324 0.9721872
35 0.0027813 1 in 360 0.9749684
36 0.0025032 1 in 400 0.9774716
37 0.0022528 1 in 444 0.9797244
38 0.0020276 1 in 494 0.9817520
39 0.0018248 1 in 549 0.9835768
40 0.0016423 1 in 609 0.9852191
41 0.0014781 1 in 677 0.9866972
42 0.0013303 1 in 752 0.9880275
43 0.0011973 1 in 836 0.9892247
44 0.0010775 1 in 929 0.9903023
45 0.0009698 1 in 1032 0.9912720
46 0.0008728 1 in 1146 0.9921448
47 0.0007855 1 in 1274 0.9929303
48 0.0007070 1 in 1415 0.9936373
49 0.0006363 1 in 1572 0.9942736
50 0.0005726 1 in 1747 0.9948462
51 0.0005154 1 in 1941 0.9953616
52 0.0004638 1 in 2156 0.9958254
53 0.0004175 1 in 2396 0.9962429
54 0.0003757 1 in 2662 0.9966186
55 0.0003381 1 in 2958 0.9969567
56 0.0003043 1 in 3286 0.9972611
57 0.0002739 1 in 3652 0.9975350
58 0.0002465 1 in 4057 0.9977815
59 0.0002219 1 in 4508 0.9980033
60 0.0001997 1 in 5009 0.9982030
61 0.0001797 1 in 5565 0.9983827
62 0.0001617 1 in 6184 0.9985444
63 0.0001456 1 in 6871 0.9986900
64 0.0001310 1 in 7634 0.9988210
65 0.0001179 1 in 8482 0.9989389
66 0.0001061 1 in 9425 0.9990450
67 0.0000955 1 in 10472 0.9991405
68 0.0000860 1 in 11635 0.9992264
69 0.0000774 1 in 12928 0.9993038
70 0.0000696 1 in 14364 0.9993734
71 0.0000627 1 in 15960 0.9994361
72 0.0000564 1 in 17733 0.9994925
73 0.0000508 1 in 19704 0.9995432
74 0.0000457 1 in 21893 0.9995889
75 0.0000411 1 in 24326 0.9996300
76 0.0000370 1 in 27028 0.9996670
77 0.0000333 1 in 30031 0.9997003
78 0.0000300 1 in 33368 0.9997303
79 0.0000270 1 in 37076 0.9997573
80 0.0000243 1 in 41195 0.9997815
81 0.0000218 1 in 45772 0.9998034
82 0.0000197 1 in 50858 0.9998230
83 0.0000177 1 in 56509 0.9998407
84 0.0000159 1 in 62788 0.9998567
85 0.0000143 1 in 69764 0.9998710
86 0.0000129 1 in 77516 0.9998839
87 0.0000116 1 in 86128 0.9998955
88 0.0000104 1 in 95698 0.9999060
89 0.0000094 1 in 106331 0.9999154
90 0.0000085 1 in 118146 0.9999238
91 0.0000076 1 in 131273 0.9999314
92 0.0000069 1 in 145859 0.9999383
93 0.0000062 1 in 162065 0.9999445
94 0.0000056 1 in 180073 0.9999500
95 0.0000050 1 in 200081 0.9999550
96 0.0000045 1 in 222312 0.9999595
97 0.0000040 1 in 247013 0.9999636
98 0.0000036 1 in 274459 0.9999672
99 0.0000033 1 in 304954 0.9999705
100 0.0000030 1 in 338838 0.9999734
101 0.0000027 1 in 376487 0.9999761
102 0.0000024 1 in 418318 0.9999785
103 0.0000022 1 in 464798 0.9999806
104 0.0000019 1 in 516442 0.9999826
105 0.0000017 1 in 573825 0.9999843

1. Random number generator; in speedrunning terms it’s become synonymous with ‘random event’ or ‘specific outcome of an event determined by [pseudo] random number generation’. I know it’s kind of weird to use “rng” as an adjective or catch-all term, but oh well. Context and such. ↩︎