DeletedUser31521
(Don't think this is truly off topic, but couldn't quite justify it as a guide. Please let me know if there is a more appropriate place for it.)
It took me a really long time to get the complete set of BPs for LoA. When I got to 45 total BPs with still one missing, I started to think that I was getting into rarified air of probability of still not having a complete set. This was by some back of the napkin math, basically guessing how likely it was to get to 8 of 9 and then saying each next BP has an 8 of 9 chance of not being the one. Something, like...
Let's assume the probability is very high you have at least 8 of 9 spots filled at 20 total BPs and n is the total number of BPs. Then you can figure the percentage of having a complete set at n greater than 20 to be at least:
equals about:
I just replace the unknown very high percentage with the highest percentage, 1. I know this isn't going to be perfect (especially at the lower value of n), but I am just looking for a rough estimate here. And I figured the likelihood of already completing the set by the time you hit 20 total BPs to offset the likelihood that my equation is bad because of rounding that probably of having at least 8 of 9 to 1.
However, I did something wrong on my initial calculation or used a starting point for very high 8 of 9 probability other than 20 or something and my results said it was way higher than 95% at 45 BPs. I also knew this was super inaccurate at just above 20 and just didn't work below 20. So... I built a program that could more accurately tell me the probabilities of having a complete set at each total number of BPs. (Because that math is actually pretty hard and I don't know how to do it.)
Code to follow, but basically I just ran a random number generator over and over to give me a number 1 through 9 (0 to 8 in computer speak). As soon as I had hit all the numbers I marked how many total attempts it took to get there (BPs) and then called it an iteration.
I tested this with the only one number I knew how to get exact with math... the probability of getting a complete set in 9 total BPs is (for reasons I won't get into):
which I was pleased to see fit with my numbers well. As you can see the numbers generally match pretty well with the earlier back of the napkin approximation math, especially when you get into the high 40's and above of total BPs. I hope this helps someone with deciding whether or not to break down and buy that final BP with diamonds. I was wrong in assuming 45 total BPs was rarified air, while 5% of the time not having it completed is pretty unlikely, I wouldn't call it "rarefied air" yet. BTW, I finally got my final BP for LoA at 58, right when I was getting to rarified air territory in my opinion. Stupid bottom right!
Anyway, I'll place my code in the next post. Sorry for the terrible formatting. I hope the mathematically and CS and just generally nerdy inclined enjoy this some. If the actual math exists and anyone can do it, I would be interested to see it!
It took me a really long time to get the complete set of BPs for LoA. When I got to 45 total BPs with still one missing, I started to think that I was getting into rarified air of probability of still not having a complete set. This was by some back of the napkin math, basically guessing how likely it was to get to 8 of 9 and then saying each next BP has an 8 of 9 chance of not being the one. Something, like...
Let's assume the probability is very high you have at least 8 of 9 spots filled at 20 total BPs and n is the total number of BPs. Then you can figure the percentage of having a complete set at n greater than 20 to be at least:
Code:
1 - [unknown very high percentage for 20 BPs equaling at least 8 sports * (8/9)^(n-20)]
equals about:
Code:
1 - [1 * (8/9)^(n-20)] --or-- 1 - (8/9)^(n-20)
I just replace the unknown very high percentage with the highest percentage, 1. I know this isn't going to be perfect (especially at the lower value of n), but I am just looking for a rough estimate here. And I figured the likelihood of already completing the set by the time you hit 20 total BPs to offset the likelihood that my equation is bad because of rounding that probably of having at least 8 of 9 to 1.
Code:
21 - 11.11% ::: 61 - 99.20%
22 - 20.99% ::: 62 - 99.29%
23 - 29.77% ::: 63 - 99.37%
24 - 37.57% ::: 64 - 99.44%
25 - 44.51% ::: 65 - 99.50%
26 - 50.67% ::: 66 - 99.56%
27 - 56.15% ::: 67 - 99.61%
28 - 61.03% ::: 68 - 99.65%
29 - 65.36% ::: 69 - 99.69%
30 - 69.21% ::: 70 - 99.72%
31 - 72.63% ::: 71 - 99.75%
32 - 75.67% ::: 72 - 99.78%
33 - 78.37% ::: 73 - 99.81%
34 - 80.78% ::: 74 - 99.83%
35 - 82.91% ::: 75 - 99.85%
36 - 84.81% ::: 76 - 99.86%
37 - 86.50% ::: 77 - 99.88%
38 - 88.00% ::: 78 - 99.89%
39 - 89.33% ::: 79 - 99.90%
40 - 90.52% ::: 80 - 99.91%
41 - 91.57% ::: 81 - 99.92%
42 - 92.51% ::: 82 - 99.93%
43 - 93.34% ::: 83 - 99.94%
44 - 94.08% ::: 84 - 99.95%
45 - 94.74% ::: 85 - 99.95%
46 - 95.32% ::: 86 - 99.96%
47 - 95.84% ::: 87 - 99.96%
48 - 96.30% ::: 88 - 99.97%
49 - 96.71% ::: 89 - 99.97%
50 - 97.08% ::: 90 - 99.97%
51 - 97.40% ::: 91 - 99.98%
52 - 97.69% ::: 92 - 99.98%
53 - 97.95% ::: 93 - 99.98%
54 - 98.18% ::: 94 - 99.98%
55 - 98.38% ::: 95 - 99.99%
56 - 98.56% ::: 96 - 99.99%
57 - 98.72% ::: 97 - 99.99%
58 - 98.86% ::: 98 - 99.99%
59 - 98.99% ::: 99 - 99.99%
60 - 99.10% ::: 100 - 99.99%
However, I did something wrong on my initial calculation or used a starting point for very high 8 of 9 probability other than 20 or something and my results said it was way higher than 95% at 45 BPs. I also knew this was super inaccurate at just above 20 and just didn't work below 20. So... I built a program that could more accurately tell me the probabilities of having a complete set at each total number of BPs. (Because that math is actually pretty hard and I don't know how to do it.)
Code to follow, but basically I just ran a random number generator over and over to give me a number 1 through 9 (0 to 8 in computer speak). As soon as I had hit all the numbers I marked how many total attempts it took to get there (BPs) and then called it an iteration.
Code:
** 10,000,000 Iterations ** ::: ** 100,000,000 Iterations **
# BPs Frequency % Complete ::: # BPs Frequency % Complete
===== ========= ========== ::: ===== ========= ==========
9 => 9343 .0934% ::: 9 => 93796 .0938%
10 => 37704 .4705% ::: 10 => 375316 .4691%
11 => 86870 1.3392% ::: 11 => 867905 1.3370%
12 => 153315 2.8723% ::: 12 => 1524115 2.8611%
13 => 227300 5.1453% ::: 13 => 2271397 5.1325%
14 => 301216 8.1575% ::: 14 => 3012284 8.1448%
15 => 367550 11.8330% ::: 15 => 3685916 11.8307%
16 => 424344 16.0764% ::: 16 => 4240844 16.0716%
17 => 466544 20.7419% ::: 17 => 4660960 20.7325%
18 => 493829 25.6802% ::: 18 => 4930878 25.6634%
19 => 509379 30.7739% ::: 19 => 5078501 30.7419%
20 => 511084 35.8848% ::: 20 => 5102089 35.8440%
21 => 502168 40.9065% ::: 21 => 5036126 40.8801%
22 => 488377 45.7902% ::: 22 => 4882002 45.7621%
23 => 467794 50.4682% ::: 23 => 4679531 50.4417%
24 => 443639 54.9046% ::: 24 => 4432260 54.8739%
25 => 415808 59.0626% ::: 25 => 4163312 59.0372%
26 => 387512 62.9378% ::: 26 => 3880981 62.9182%
27 => 357311 66.5109% ::: 27 => 3587469 66.5057%
28 => 330815 69.8190% ::: 28 => 3302584 69.8083%
29 => 302083 72.8399% ::: 29 => 3023810 72.8321%
30 => 275686 75.5967% ::: 30 => 2762274 75.5944%
31 => 250923 78.1059% ::: 31 => 2507865 78.1022%
32 => 226990 80.3758% ::: 32 => 2274649 80.3769%
33 => 205785 82.4337% ::: 33 => 2058295 82.4352%
34 => 184829 84.2820% ::: 34 => 1848730 84.2839%
35 => 166800 85.9500% ::: 35 => 1670094 85.9540%
36 => 149420 87.4442% ::: 36 => 1499929 87.4539%
37 => 134454 88.7887% ::: 37 => 1349558 88.8035%
38 => 121239 90.0011% ::: 38 => 1208533 90.0120%
39 => 107896 91.0801% ::: 39 => 1082951 91.0950%
40 => 96662 92.0467% ::: 40 => 967299 92.0623%
41 => 86843 92.9151% ::: 41 => 863231 92.9255%
42 => 77821 93.6933% ::: 42 => 773471 93.6990%
43 => 68747 94.3808% ::: 43 => 688014 94.3870%
44 => 62076 95.0016% ::: 44 => 614293 95.0013%
45 => 54811 95.5497% ::: 45 => 550087 95.5513%
46 => 49076 96.0404% ::: 46 => 489149 96.0405%
47 => 43481 96.4752% ::: 47 => 436434 96.4769%
48 => 38885 96.8641% ::: 48 => 388601 96.8655%
49 => 34512 97.2092% ::: 49 => 346636 97.2122%
50 => 30918 97.5184% ::: 50 => 307966 97.5201%
51 => 27352 97.7919% ::: 51 => 274311 97.7944%
52 => 24179 98.0337% ::: 52 => 243899 98.0383%
53 => 22079 98.2545% ::: 53 => 217362 98.2557%
54 => 19320 98.4477% ::: 54 => 192804 98.4485%
55 => 17348 98.6212% ::: 55 => 173644 98.6222%
56 => 15286 98.7740% ::: 56 => 152395 98.7746%
57 => 13529 98.9093% ::: 57 => 136051 98.9106%
58 => 12289 99.0322% ::: 58 => 120163 99.0308%
59 => 10687 99.1391% ::: 59 => 107023 99.1378%
60 => 9461 99.2337% ::: 60 => 95967 99.2338%
61 => 8559 99.3193% ::: 61 => 84811 99.3186%
62 => 7547 99.3948% ::: 62 => 75247 99.3938%
63 => 6829 99.4630% ::: 63 => 66910 99.4607%
64 => 5950 99.5225% ::: 64 => 59614 99.5203%
65 => 5217 99.5747% ::: 65 => 52820 99.5732%
66 => 4676 99.6215% ::: 66 => 47346 99.6205%
67 => 4188 99.6634% ::: 67 => 42137 99.6626%
68 => 3942 99.7028% ::: 68 => 39249 99.7019%
69 => 3229 99.7351% ::: 69 => 33071 99.7350%
70 => 2892 99.7640% ::: 70 => 29367 99.7643%
71 => 2709 99.7911% ::: 71 => 26191 99.7905%
72 => 2289 99.8140% ::: 72 => 23164 99.8137%
73 => 2005 99.8340% ::: 73 => 20669 99.8344%
74 => 1739 99.8514% ::: 74 => 18269 99.8526%
75 => 1662 99.8680% ::: 75 => 16363 99.8690%
76 => 1471 99.8827% ::: 76 => 14584 99.8836%
77 => 1303 99.8958% ::: 77 => 12847 99.8964%
78 => 1119 99.9070% ::: 78 => 11451 99.9079%
79 => 1017 99.9171% ::: 79 => 10173 99.9180%
80 => 910 99.9262% ::: 80 => 8974 99.9270%
81 => 790 99.9341% ::: 81 => 8031 99.9350%
82 => 747 99.9416% ::: 82 => 7155 99.9422%
83 => 628 99.9479% ::: 83 => 6324 99.9485%
84 => 569 99.9536% ::: 84 => 5814 99.9543%
85 => 514 99.9587% ::: 85 => 5108 99.9594%
86 => 428 99.9630% ::: 86 => 4495 99.9639%
87 => 411 99.9671% ::: 87 => 4098 99.9680%
88 => 372 99.9708% ::: 88 => 3476 99.9715%
89 => 314 99.9740% ::: 89 => 3041 99.9746%
90 => 285 99.9768% ::: 90 => 2777 99.9773%
91 => 260 99.9794% ::: 91 => 2494 99.9798%
92 => 224 99.9816% ::: 92 => 2227 99.9821%
93 => 180 99.9834% ::: 93 => 1914 99.9840%
94 => 184 99.9853% ::: 94 => 1726 99.9857%
95 => 157 99.9869% ::: 95 => 1566 99.9873%
96 => 132 99.9882% ::: 96 => 1438 99.9887%
97 => 154 99.9897% ::: 97 => 1260 99.9900%
98 => 117 99.9909% ::: 98 => 1100 99.9911%
99 => 104 99.9919% ::: 99 => 964 99.9920%
100 => 76 99.9927% ::: 100 => 858 99.9929%
101 => 95 99.9936% ::: 101 => 787 99.9937%
102 => 83 99.9945% ::: 102 => 670 99.9943%
103 => 70 99.9952% ::: 103 => 618 99.9950%
104 => 67 99.9958% ::: 104 => 558 99.9955%
105 => 42 99.9963% ::: 105 => 505 99.9960%
106 => 43 99.9967% ::: 106 => 442 99.9965%
107 => 33 99.9970% ::: 107 => 385 99.9968%
108 => 37 99.9974% ::: 108 => 355 99.9972%
109 => 25 99.9976% ::: 109 => 308 99.9975%
110 => 26 99.9979% ::: 110 => 283 99.9978%
111 => 22 99.9981% ::: 111 => 242 99.9980%
112 => 23 99.9983% ::: 112 => 216 99.9982%
113 => 23 99.9986% ::: 113 => 170 99.9984%
114 => 15 99.9987% ::: 114 => 162 99.9986%
115 => 17 99.9989% ::: 115 => 167 99.9987%
116 => 11 99.9990% ::: 116 => 139 99.9989%
117 => 6 99.9991% ::: 117 => 125 99.9990%
118 => 9 99.9992% ::: 118 => 100 99.9991%
119 => 7 99.9992% ::: 119 => 101 99.9992%
120 => 8 99.9993% ::: 120 => 80 99.9993%
121 => 8 99.9994% ::: 121 => 80 99.9994%
122 => 6 99.9994% ::: 122 => 71 99.9994%
123 => 4 99.9995% ::: 123 => 76 99.9995%
124 => 7 99.9996% ::: 124 => 70 99.9996%
125 => 2 99.9996% ::: 125 => 44 99.9996%
126 => 2 99.9996% ::: 126 => 42 99.9997%
127 => 6 99.9997% ::: 127 => 32 99.9997%
128 => 4 99.9997% ::: 128 => 34 99.9997%
129 => 3 99.9997% ::: 129 => 35 99.9998%
130 => 4 99.9998% ::: 130 => 26 99.9998%
131 => 4 99.9998% ::: 131 => 19 99.9998%
132 => 3 99.9998% ::: 132 => 20 99.9998%
133 => 2 99.9999% ::: 133 => 21 99.9999%
135 => 3 99.9999% ::: 134 => 10 99.9999%
136 => 1 99.9999% ::: 135 => 19 99.9999%
137 => 2 99.9999% ::: 136 => 9 99.9999%
138 => 1 99.9999% ::: 137 => 10 99.9999%
141 => 2 99.9999% ::: 138 => 12 99.9999%
144 => 1 100.0000% ::: 139 => 9 99.9999%
148 => 2 100.0000% ::: 140 => 4 99.9999%
150 => 1 100.0000% ::: 141 => 13 99.9999%
158 => 1 100.0000% ::: 142 => 11 100.0000%
162 => 1 100.0000% ::: 143 => 6 100.0000%
::: 144 => 4 100.0000%
::: 145 => 2 100.0000%
::: 146 => 3 100.0000%
::: 147 => 3 100.0000%
::: 148 => 3 100.0000%
::: 149 => 3 100.0000%
::: 150 => 1 100.0000%
::: 151 => 2 100.0000%
::: 152 => 1 100.0000%
::: 153 => 1 100.0000%
::: 154 => 2 100.0000%
::: 155 => 3 100.0000%
::: 156 => 2 100.0000%
::: 157 => 2 100.0000%
::: 158 => 3 100.0000%
::: 168 => 1 100.0000%
::: 179 => 1 100.0000%
I tested this with the only one number I knew how to get exact with math... the probability of getting a complete set in 9 total BPs is (for reasons I won't get into):
Code:
9! / (9^9) = .09366567%
which I was pleased to see fit with my numbers well. As you can see the numbers generally match pretty well with the earlier back of the napkin approximation math, especially when you get into the high 40's and above of total BPs. I hope this helps someone with deciding whether or not to break down and buy that final BP with diamonds. I was wrong in assuming 45 total BPs was rarified air, while 5% of the time not having it completed is pretty unlikely, I wouldn't call it "rarefied air" yet. BTW, I finally got my final BP for LoA at 58, right when I was getting to rarified air territory in my opinion. Stupid bottom right!
Anyway, I'll place my code in the next post. Sorry for the terrible formatting. I hope the mathematically and CS and just generally nerdy inclined enjoy this some. If the actual math exists and anyone can do it, I would be interested to see it!