Satoshi Dice -- Statistical Analysis

[Stephen Gornick]

I must be one unlucky bastard...

Well, unlike games of chance where you need to trust a machine or a human, SatoshiDICE's results are verifiable, thanks to cryptography, once the secret is released (the next day).

This can be verified without needing any software installed.  Here's how:

So for your wagers from Jul 10, 2012 (UTC), get the secret from:
 - http://satoshidice.com/secretlist.php

That has been released, and it is:
  ctygmtZjw7xTnLou7u9cWVYiOd79xAMJK5aJJBBD3w2Su8vGMHPDwP1f4O1eBPek

Then create the SHA-256 hash of that secret with:
 - http://md5online.eu/3,sha-256

Which yields:
 - d75e37bc91915af566276e34624e7477c2a6771fb7898bc4dd1b30ccc0022cfe

Then verify that the resulting hash from that jives with the has from hash.keys for Jul 10, 2012:
 - https://s3-eu-west-1.amazonaws.com/satoshidice/hash.keys

So next you grab the transaction ID for your transaction (from the Transactions tab of Bitcoin.org client).

Then plunk those values in ... for Text to convert, paste the Transaction ID, and for Shared Secret Key use the secret for the day that you verified above.

 - http://www.online-convert.com/result/3a4827ec61a06ed23ce463afd509c4ce

After you click Calculate hash, you'll get a hex string in the results.  The first four alphanumeric characters is the hex representation of your lucky number (from 0000 through ffff hex, or in decimal, 0 through 65,535 decimal).  You can convert those values to decimal here:

 - http://www.binaryhexconverter.com/hex-to-decimal-converter

So if you had bet the "less than 8,000" wager and your lucky number was 8,000 or higher, ... that specific wager didn't win.

As long as the hashes to the secrets are known in advance (and they are, that hash.keys was published in April, a copy of which has been archived here) you can always verify if you should have gotten a winning payout or not:
 - https://raw.github.com/sgornick/satoshidice/master/hash.keys

[1713428592]

1713428592

[1713428592]

1713428592

[1713428592]

1713428592

[dooglus]

I've spent all day re-downloading the blockchain after a hard drive failure corrupted my copy.  Here's an update to the SatoshiDice stats.  Looks like the site is up over 1000 BTC since the last time I posted.

Results: 2012-Jul-11 02:33am (up to block 188553)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |     7243 |      0 (0.00000) |   6981 |     262 |      28.11 |       0.01 |    15.91 |     28.09 |   0.068
 1dice1Qf4       2      0.00003 |      958 |      0 (0.00000) |    889 |      69 |       7.41 |       0.00 |     5.38 |      7.41 |   0.024
 1dice2pxm       4      0.00006 |     1457 |      0 (0.00000) |   1426 |      31 |      12.65 |       0.00 |     1.22 |     12.65 |   0.044
 1dice2vQo       8      0.00012 |     1226 |      0 (0.00000) |   1188 |      38 |      17.18 |       0.00 |     3.15 |     17.18 |   0.028
 1dice2WmR      16      0.00024 |     1422 |      0 (0.00000) |   1394 |      28 |      22.18 |       0.01 |     4.95 |     22.16 |   0.076
 1dice2xkj      32      0.00049 |     3197 |      1 (0.00031) |   3185 |      11 |      96.95 |     100.40 |     1.29 |     -3.44 | 103.553
 1dice2zdo      64      0.00098 |     4746 |      6 (0.00127) |   4723 |      17 |     189.63 |     120.67 |    55.64 |     68.96 |  63.633
 1dice37Ee     128      0.00195 |     6037 |     12 (0.00200) |   5978 |      47 |    1221.85 |    1142.24 |    39.25 |     79.61 |  93.484
 1dice3jkp     256      0.00391 |     4487 |     21 (0.00469) |   4454 |      12 |     469.95 |     331.94 |    11.11 |    138.00 |  70.634
 1dice4J1m     512      0.00781 |     6106 |     35 (0.00574) |   6066 |       5 |    1194.44 |     346.61 |     9.35 |    847.82 |  29.019
 1dice5wwE    1000      0.01526 |     8142 |    110 (0.01351) |   8030 |       2 |    1654.70 |     848.15 |     1.80 |    806.54 |  51.257
 1dice61SN    1500      0.02289 |     6489 |    146 (0.02252) |   6337 |       6 |    2725.20 |    3010.39 |    15.00 |   -285.19 | 110.465
 1dice6DPt    2000      0.03052 |     6873 |    209 (0.03042) |   6661 |       3 |    2670.04 |    2129.62 |     9.24 |    540.42 |  79.760
 1dice6gJg    3000      0.04578 |     6397 |    309 (0.04836) |   6081 |       7 |    4446.51 |    5562.59 |    24.99 |  -1116.08 | 125.100
 1dice6GV5    4000      0.06104 |     6694 |    419 (0.06262) |   6272 |       3 |    2292.38 |    2030.50 |    31.20 |    261.87 |  88.576
 1dice6wBx    6000      0.09155 |    12181 |   1153 (0.09469) |  11023 |       5 |    7201.29 |    7611.02 |     7.01 |   -409.72 | 105.690
 1dice6YgE    8000      0.12207 |    22679 |   2807 (0.12377) |  19872 |       0 |    4396.67 |    3957.72 |     0.00 |    438.94 |  90.016
 1dice7EYz   12000      0.18311 |    15757 |   2996 (0.19020) |  12756 |       5 |    6437.21 |    6701.99 |    14.50 |   -264.78 | 104.113
 1dice7fUk   16000      0.24414 |    38674 |   9347 (0.24173) |  29320 |       7 |   11004.63 |   10455.74 |    97.79 |    548.89 |  95.012
 1dice7W2A   24000      0.36621 |    28312 |  10452 (0.36958) |  17829 |      31 |   12506.80 |   12496.78 |   212.63 |     10.02 |  99.920
 1dice8EMZ   32000      0.48828 |   283521 | 138117 (0.48736) | 145279 |     125 |   84697.01 |   86053.02 |  2173.21 |  -1356.01 | 101.601
 1dice97EC   32768      0.50000 |   118103 |  58967 (0.49957) |  59069 |      67 |   41961.41 |   40734.94 |   789.19 |   1226.46 |  97.077
 1dice9wcM   48000      0.73242 |    71043 |  52202 (0.73522) |  18800 |      41 |   41553.51 |   40079.90 |   442.88 |   1473.61 |  96.454
 1dice9wVt   64000      0.97656 |     5697 |   4598 (0.97457) |    120 |     979 |    4901.48 |    4709.70 |   287.60 |    191.78 |  96.087
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   667441 | 281907           | 383733 |    1801 |  231709.32 |  228424.06 |  4254.38 |   3285.25 |  98.582
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3285.25987455 BTC (1.418%)
Cumulative Fees Paid:         335.89477500 BTC
SD Profit after fees:        2949.36509955 BTC (1.273%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  1975820  :  SatoshiDice Tx: 1239235  (62.7%)
Blockchain MB:  835.1  :  SatoshiDice Tx: 505.3  (60.5%)

[dooglus]

I fixed a bug in the analysis script that was misclassifying some small "lessthan 64000" and "lessthan 48000" wins as refunds.

Results: 2012-Jul-11 12:27pm (up to block 188606)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |     7311 |      0 (0.00000) |   7048 |     263 |      28.32 |       0.01 |    16.01 |     28.30 |   0.068
 1dice1Qf4       2      0.00003 |      960 |      0 (0.00000) |    891 |      69 |       7.43 |       0.00 |     5.38 |      7.43 |   0.024
 1dice2pxm       4      0.00006 |     1460 |      0 (0.00000) |   1429 |      31 |      12.68 |       0.00 |     1.22 |     12.68 |   0.044
 1dice2vQo       8      0.00012 |     1228 |      0 (0.00000) |   1190 |      38 |      17.20 |       0.00 |     3.15 |     17.20 |   0.028
 1dice2WmR      16      0.00024 |     1426 |      0 (0.00000) |   1397 |      29 |      22.21 |       0.01 |     5.85 |     22.19 |   0.076
 1dice2xkj      32      0.00049 |     3201 |      1 (0.00031) |   3189 |      11 |      96.99 |     100.40 |     1.29 |     -3.40 | 103.510
 1dice2zdo      64      0.00098 |     4952 |      6 (0.00122) |   4929 |      17 |     194.29 |     120.67 |    55.64 |     73.62 |  62.106
 1dice37Ee     128      0.00195 |     6049 |     13 (0.00217) |   5989 |      47 |    1222.67 |    1142.73 |    39.25 |     79.94 |  93.462
 1dice3jkp     256      0.00391 |     4490 |     21 (0.00469) |   4457 |      12 |     470.16 |     331.94 |    11.11 |    138.21 |  70.602
 1dice4J1m     512      0.00781 |     6214 |     37 (0.00596) |   6172 |       5 |    1196.30 |     354.02 |     9.35 |    842.27 |  29.593
 1dice5wwE    1000      0.01526 |     8235 |    112 (0.01360) |   8121 |       2 |    1664.95 |     854.50 |     1.80 |    810.44 |  51.323
 1dice61SN    1500      0.02289 |     6507 |    148 (0.02277) |   6353 |       6 |    2726.25 |    3014.61 |    15.00 |   -288.36 | 110.577
 1dice6DPt    2000      0.03052 |     6940 |    215 (0.03099) |   6722 |       3 |    2679.79 |    2151.78 |     9.24 |    528.01 |  80.296
 1dice6gJg    3000      0.04578 |     6412 |    311 (0.04856) |   6094 |       7 |    4447.46 |    5564.70 |    24.99 |  -1117.24 | 125.121
 1dice6GV5    4000      0.06104 |     6717 |    422 (0.06285) |   6292 |       3 |    2292.97 |    2032.27 |    31.20 |    260.70 |  88.630
 1dice6wBx    6000      0.09155 |    12411 |   1167 (0.09407) |  11239 |       5 |    7229.29 |    7636.41 |     7.01 |   -407.12 | 105.632
 1dice6YgE    8000      0.12207 |    22857 |   2826 (0.12364) |  20031 |       0 |    4411.04 |    3968.68 |     0.00 |    442.35 |  89.972
 1dice7EYz   12000      0.18311 |    15796 |   3000 (0.18998) |  12791 |       5 |    6458.12 |    6719.99 |    14.50 |   -261.87 | 104.055
 1dice7fUk   16000      0.24414 |    38932 |   9410 (0.24175) |  29515 |       7 |   11130.51 |   10567.06 |    97.79 |    563.44 |  94.938
 1dice7W2A   24000      0.36621 |    28408 |  10490 (0.36967) |  17887 |      31 |   12540.05 |   12526.28 |   212.63 |     13.76 |  99.890
 1dice8EMZ   32000      0.48828 |   283749 | 138222 (0.48734) | 145402 |     125 |   84751.02 |   86109.84 |  2173.21 |  -1358.81 | 101.603
 1dice97EC   32768      0.50000 |   118730 |  59259 (0.49939) |  59403 |      68 |   42161.69 |   40927.90 |   789.19 |   1233.79 |  97.074
 1dice9wcM   48000      0.73242 |    71404 |  52477 (0.73534) |  18887 |      40 |   41687.34 |   40205.86 |   442.88 |   1481.47 |  96.446
 1dice9wVt   64000      0.97656 |     5698 |   5452 (0.97846) |    120 |     126 |    4952.80 |    4761.02 |   239.20 |    191.78 |  96.128
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   670087 | 283589           | 385548 |     950 |  232401.64 |  229090.79 |  4206.99 |   3310.85 |  98.575
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3310.85104294 BTC (1.425%)
Cumulative Fees Paid:         337.22142500 BTC
SD Profit after fees:        2973.62961794 BTC (1.280%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  1985935  :  SatoshiDice Tx: 1243858  (62.6%)
Blockchain MB:  839.2  :  SatoshiDice Tx: 507.2  (60.4%)

[dooglus]

Results: 2012-Jul-12 10:43pm (up to block 188813)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |     7484 |      0 (0.00000) |   7220 |     264 |      29.45 |       0.01 |    16.06 |     29.44 |   0.065
 1dice1Qf4       2      0.00003 |      972 |      0 (0.00000) |    903 |      69 |       7.56 |       0.00 |     5.38 |      7.56 |   0.023
 1dice2pxm       4      0.00006 |     1482 |      0 (0.00000) |   1450 |      32 |      12.98 |       0.00 |     1.22 |     12.98 |   0.043
 1dice2vQo       8      0.00012 |     1251 |      0 (0.00000) |   1213 |      38 |      17.41 |       0.00 |     3.15 |     17.41 |   0.027
 1dice2WmR      16      0.00024 |     1447 |      0 (0.00000) |   1418 |      29 |      22.45 |       0.01 |     5.85 |     22.43 |   0.075
 1dice2xkj      32      0.00049 |     3241 |      1 (0.00031) |   3229 |      11 |      98.75 |     100.40 |     1.29 |     -1.64 | 101.670
 1dice2zdo      64      0.00098 |     5036 |      6 (0.00120) |   5013 |      17 |     195.18 |     120.67 |    55.64 |     74.51 |  61.823
 1dice37Ee     128      0.00195 |     6063 |     13 (0.00216) |   6003 |      47 |    1222.78 |    1142.73 |    39.25 |     80.04 |  93.454
 1dice3jkp     256      0.00391 |     4498 |     21 (0.00468) |   4464 |      13 |     470.80 |     331.94 |    13.11 |    138.85 |  70.507
 1dice4J1m     512      0.00781 |     6437 |     40 (0.00622) |   6392 |       5 |    1234.44 |     455.44 |     9.35 |    778.99 |  36.895
 1dice5wwE    1000      0.01526 |     8586 |    121 (0.01410) |   8463 |       2 |    1706.56 |     959.10 |     1.80 |    747.45 |  56.201
 1dice61SN    1500      0.02289 |     6667 |    154 (0.02312) |   6507 |       6 |    2764.38 |    3083.46 |    15.00 |   -319.08 | 111.543
 1dice6DPt    2000      0.03052 |     7146 |    227 (0.03178) |   6916 |       3 |    2814.97 |    2555.55 |     9.24 |    259.42 |  90.784
 1dice6gJg    3000      0.04578 |     6599 |    325 (0.04930) |   6267 |       7 |    4509.55 |    5658.32 |    24.99 |  -1148.77 | 125.474
 1dice6GV5    4000      0.06104 |     6981 |    445 (0.06377) |   6533 |       3 |    2374.74 |    2160.17 |    31.20 |    214.57 |  90.964
 1dice6wBx    6000      0.09155 |    12908 |   1223 (0.09478) |  11680 |       5 |    7470.22 |    7897.49 |     7.01 |   -427.26 | 105.720
 1dice6YgE    8000      0.12207 |    23642 |   2939 (0.12431) |  20703 |       0 |    4772.73 |    4349.05 |     0.00 |    423.67 |  91.123
 1dice7EYz   12000      0.18311 |    15863 |   3008 (0.18968) |  12850 |       5 |    6534.98 |    6726.51 |    14.50 |   -191.52 | 102.931
 1dice7fUk   16000      0.24414 |    39988 |   9687 (0.24229) |  30294 |       7 |   12138.42 |   11810.53 |    97.79 |    327.88 |  97.299
 1dice7W2A   24000      0.36621 |    28911 |  10703 (0.37060) |  18177 |      31 |   12769.01 |   12821.37 |   212.63 |    -52.36 | 100.410
 1dice8EMZ   32000      0.48828 |   285795 | 139241 (0.48742) | 146428 |     126 |   86878.31 |   88174.73 |  2173.21 |  -1296.41 | 101.492
 1dice97EC   32768      0.50000 |   119763 |  59761 (0.49928) |  59934 |      68 |   42500.15 |   41349.33 |   789.19 |   1150.82 |  97.292
 1dice9wcM   48000      0.73242 |    72117 |  52999 (0.73533) |  19076 |      42 |   42130.05 |   40653.69 |   467.98 |   1476.36 |  96.496
 1dice9wVt   64000      0.97656 |     5698 |   5452 (0.97846) |    120 |     126 |    4952.80 |    4761.02 |   239.20 |    191.78 |  96.128
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   678575 | 286366           | 391253 |     956 |  237628.79 |  235111.64 |  4234.14 |   2517.15 |  98.941
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       2517.15337759 BTC (1.059%)
Cumulative Fees Paid:         341.55510000 BTC
SD Profit after fees:        2175.59827759 BTC (0.916%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2022603  :  SatoshiDice Tx: 1258148  (62.2%)
Blockchain MB:  854.8  :  SatoshiDice Tx: 513.5  (60.1%)

[unclemantis]

Tried it a few times last night. Even picked the higher numbers. Lost every time.

What is the pick range of the magic number?

[dooglus]

Tried it a few times last night. Even picked the higher numbers. Lost every time.

What is the pick range of the magic number?

It's 0 through 65535 inclusive.

So when you play "lessthan 48000" you should win 73.2% of the time, and when you play "lessthan 64000" you should win 97.6% of the time.

I played lessthan 48000 3 times yesterday and only won once (luckily biggest of the 3 bets was the one that won).  But the games are proveably fair - you can get the secret number they used to pick your magic number and validate that they did the maths correctly.

One guy was saying he lost over 60 "lessthan 8000" games in a row the other day, so long losing streaks are definitely possible, just very unlikely.

[unclemantis]

Tried it a few times last night. Even picked the higher numbers. Lost every time.

What is the pick range of the magic number?

It's 0 through 65535 inclusive.

So when you play "lessthan 48000" you should win 73.2% of the time, and when you play "lessthan 64000" you should win 97.6% of the time.

I played lessthan 48000 3 times yesterday and only won once (luckily biggest of the 3 bets was the one that won).  But the games are proveably fair - you can get the secret number they used to pick your magic number and validate that they did the maths correctly.

One guy was saying he lost over 60 "lessthan 8000" games in a row the other day, so long losing streaks are definitely possible, just very unlikely.

I don't have very good luck with lotteries. Does the same carry over into bitcoin? :-P

[dooglus]

I don't have very good luck with lotteries. Does the same carry over into bitcoin? :-P

Maybe.  Lotteries usually have really bad house edges (around 50% or so), whereas SatoshiDice only currently takes 1.9%.  So you're less likely to be unlucky at SatoshiDice, mathematically speaking.  But when the flying spaghetti monster is out to get you, mathematical odds don't matter.

Results: 2012-Jul-15 09:53am (up to block 189202)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |     8352 |      0 (0.00000) |   8087 |     265 |      35.66 |       0.01 |    16.06 |     35.64 |   0.054
 1dice1Qf4       2      0.00003 |      974 |      0 (0.00000) |    905 |      69 |       7.59 |       0.00 |     5.38 |      7.59 |   0.023
 1dice2pxm       4      0.00006 |     1487 |      0 (0.00000) |   1455 |      32 |      13.03 |       0.00 |     1.22 |     13.03 |   0.043
 1dice2vQo       8      0.00012 |     1253 |      0 (0.00000) |   1215 |      38 |      17.43 |       0.00 |     3.15 |     17.43 |   0.027
 1dice2WmR      16      0.00024 |     1461 |      0 (0.00000) |   1431 |      30 |      22.78 |       0.01 |     6.60 |     22.76 |   0.074
 1dice2xkj      32      0.00049 |     3369 |      1 (0.00030) |   3357 |      11 |     105.63 |     100.40 |     1.29 |      5.23 |  95.047
 1dice2zdo      64      0.00098 |     5122 |      6 (0.00118) |   5099 |      17 |     205.03 |     120.68 |    55.64 |     84.34 |  58.861
 1dice37Ee     128      0.00195 |     6147 |     13 (0.00213) |   6086 |      48 |    1231.82 |    1142.74 |    40.25 |     89.08 |  92.768
 1dice3jkp     256      0.00391 |     4544 |     21 (0.00463) |   4510 |      13 |     486.66 |     332.01 |    13.11 |    154.65 |  68.222
 1dice4J1m     512      0.00781 |     6736 |     42 (0.00624) |   6689 |       5 |    1451.40 |     580.08 |     9.35 |    871.32 |  39.967
 1dice5wwE    1000      0.01526 |     9332 |    137 (0.01468) |   9193 |       2 |    1965.26 |    1545.90 |     1.80 |    419.35 |  78.661
 1dice61SN    1500      0.02289 |     6879 |    160 (0.02328) |   6713 |       6 |    2892.46 |    3248.44 |    15.00 |   -355.97 | 112.307
 1dice6DPt    2000      0.03052 |     7475 |    240 (0.03212) |   7232 |       3 |    3198.67 |    2892.58 |     9.24 |    306.09 |  90.431
 1dice6gJg    3000      0.04578 |     6826 |    343 (0.05030) |   6476 |       7 |    4752.43 |    6300.69 |    24.99 |  -1548.25 | 132.578
 1dice6GV5    4000      0.06104 |     7305 |    465 (0.06368) |   6837 |       3 |    2678.50 |    2424.83 |    31.20 |    253.66 |  90.530
 1dice6wBx    6000      0.09155 |    13901 |   1302 (0.09370) |  12594 |       5 |    8452.17 |    8638.28 |     7.01 |   -186.10 | 102.202
 1dice6YgE    8000      0.12207 |    25985 |   3223 (0.12403) |  22762 |       0 |    5892.84 |    5184.56 |     0.00 |    708.28 |  87.981
 1dice7EYz   12000      0.18311 |    15912 |   3015 (0.18954) |  12892 |       5 |    6562.31 |    6727.49 |    14.50 |   -165.18 | 102.517
 1dice7fUk   16000      0.24414 |    41160 |   9972 (0.24232) |  31181 |       7 |   12896.17 |   12536.86 |    97.79 |    359.30 |  97.214
 1dice7W2A   24000      0.36621 |    30158 |  11162 (0.37050) |  18965 |      31 |   12969.95 |   13008.20 |   212.63 |    -38.25 | 100.295
 1dice8EMZ   32000      0.48828 |   288217 | 140427 (0.48744) | 147664 |     126 |   88105.16 |   89311.03 |  2173.21 |  -1205.87 | 101.369
 1dice97EC   32768      0.50000 |   121162 |  60435 (0.49908) |  60658 |      69 |   42889.19 |   41626.60 |   789.19 |   1262.58 |  97.056
 1dice9wcM   48000      0.73242 |    76260 |  56098 (0.73602) |  20120 |      42 |   46421.48 |   44908.58 |   467.98 |   1512.90 |  96.741
 1dice9wVt   64000      0.97656 |     5728 |   5480 (0.97840) |    121 |     127 |    4965.69 |    4773.70 |   239.20 |    191.99 |  96.134
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   695745 | 292542           | 402242 |     961 |  248219.43 |  245403.78 |  4235.89 |   2815.65 |  98.866
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       2815.65385948 BTC (1.134%)
Cumulative Fees Paid:         350.27962500 BTC
SD Profit after fees:        2465.37423448 BTC (0.993%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2084393  :  SatoshiDice Tx: 1286610  (61.7%)
Blockchain MB:  881.0  :  SatoshiDice Tx: 526.3  (59.7%)

[dooglus]

Results: 2012-Jul-16 11:29pm (up to block 189448)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |     8718 |      0 (0.00000) |   8450 |     268 |      37.07 |       0.01 |    16.20 |     37.05 |   0.052
 1dice1Qf4       2      0.00003 |      975 |      0 (0.00000) |    906 |      69 |       7.61 |       0.00 |     5.38 |      7.61 |   0.023
 1dice2pxm       4      0.00006 |     1487 |      0 (0.00000) |   1455 |      32 |      13.03 |       0.00 |     1.22 |     13.03 |   0.043
 1dice2vQo       8      0.00012 |     1259 |      0 (0.00000) |   1220 |      39 |      17.48 |       0.00 |     3.65 |     17.48 |   0.027
 1dice2WmR      16      0.00024 |     1463 |      0 (0.00000) |   1433 |      30 |      22.78 |       0.01 |     6.60 |     22.77 |   0.074
 1dice2xkj      32      0.00049 |     3374 |      1 (0.00030) |   3362 |      11 |     105.67 |     100.40 |     1.29 |      5.27 |  95.010
 1dice2zdo      64      0.00098 |     5140 |      7 (0.00137) |   5116 |      17 |     206.31 |     121.68 |    55.64 |     84.62 |  58.983
 1dice37Ee     128      0.00195 |     6231 |     14 (0.00226) |   6169 |      48 |    1232.64 |    1143.24 |    40.25 |     89.39 |  92.747
 1dice3jkp     256      0.00391 |     4553 |     21 (0.00463) |   4519 |      13 |     489.69 |     332.02 |    13.11 |    157.67 |  67.802
 1dice4J1m     512      0.00781 |     6740 |     42 (0.00624) |   6693 |       5 |    1451.41 |     580.08 |     9.35 |    871.33 |  39.966
 1dice5wwE    1000      0.01526 |     9686 |    142 (0.01466) |   9542 |       2 |    2004.63 |    1583.13 |     1.80 |    421.49 |  78.974
 1dice61SN    1500      0.02289 |     6886 |    160 (0.02326) |   6720 |       6 |    2900.48 |    3248.48 |    15.00 |   -347.99 | 111.998
 1dice6DPt    2000      0.03052 |     7483 |    240 (0.03209) |   7240 |       3 |    3198.76 |    2892.58 |     9.24 |    306.18 |  90.428
 1dice6gJg    3000      0.04578 |     6832 |    343 (0.05026) |   6482 |       7 |    4752.59 |    6300.69 |    24.99 |  -1548.09 | 132.574
 1dice6GV5    4000      0.06104 |     7310 |    467 (0.06391) |   6840 |       3 |    2678.86 |    2428.34 |    31.20 |    250.52 |  90.648
 1dice6wBx    6000      0.09155 |    14039 |   1311 (0.09342) |  12723 |       5 |    8484.50 |    8643.82 |     7.01 |   -159.31 | 101.878
 1dice6YgE    8000      0.12207 |    26582 |   3305 (0.12433) |  23277 |       0 |    5920.44 |    5201.16 |     0.00 |    719.28 |  87.851
 1dice7EYz   12000      0.18311 |    15966 |   3026 (0.18959) |  12935 |       5 |    6580.53 |    6729.73 |    14.50 |   -149.20 | 102.267
 1dice7fUk   16000      0.24414 |    41425 |  10031 (0.24219) |  31387 |       7 |   12993.44 |   12583.89 |    97.79 |    409.54 |  96.848
 1dice7W2A   24000      0.36621 |    30709 |  11370 (0.37064) |  19307 |      32 |   13071.08 |   13064.08 |   212.63 |      6.99 |  99.946
 1dice8EMZ   32000      0.48828 |   290403 | 141496 (0.48745) | 148781 |     126 |   88791.93 |   89952.93 |  2173.21 |  -1160.99 | 101.308
 1dice97EC   32768      0.50000 |   122044 |  60876 (0.49911) |  61094 |      74 |   43393.84 |   42090.60 |   789.20 |   1303.24 |  96.997
 1dice9wcM   48000      0.73242 |    83554 |  61462 (0.73597) |  22050 |      42 |   51177.30 |   49602.31 |   467.98 |   1574.99 |  96.922
 1dice9wVt   64000      0.97656 |     5753 |   5504 (0.97831) |    122 |     127 |    5005.53 |    4812.71 |   239.20 |    192.82 |  96.148
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   708612 | 299818           | 407823 |     971 |  254537.74 |  251411.99 |  4236.52 |   3125.75 |  98.772
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3125.75005544 BTC (1.228%)
Cumulative Fees Paid:         356.72815000 BTC
SD Profit after fees:        2769.02190544 BTC (1.088%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2132971  :  SatoshiDice Tx: 1310999  (61.5%)
Blockchain MB:  902.0  :  SatoshiDice Tx: 537.1  (59.5%)

[dooglus]

Results: 2012-Jul-17 11:39am (up to block 189537)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP  
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |     8980 |      0 (0.00000) |   8712 |     268 |      38.08 |       0.01 |    16.20 |     38.06 |   0.051
 1dice1Qf4       2      0.00003 |      976 |      0 (0.00000) |    907 |      69 |       7.71 |       0.00 |     5.38 |      7.71 |   0.023
 1dice2pxm       4      0.00006 |     1487 |      0 (0.00000) |   1455 |      32 |      13.03 |       0.00 |     1.22 |     13.03 |   0.043
 1dice2vQo       8      0.00012 |     1259 |      0 (0.00000) |   1220 |      39 |      17.48 |       0.00 |     3.65 |     17.48 |   0.027
 1dice2WmR      16      0.00024 |     1463 |      0 (0.00000) |   1433 |      30 |      22.78 |       0.01 |     6.60 |     22.77 |   0.074
 1dice2xkj      32      0.00049 |     3382 |      1 (0.00030) |   3370 |      11 |     105.75 |     100.40 |     1.29 |      5.35 |  94.938
 1dice2zdo      64      0.00098 |     5142 |      7 (0.00137) |   5118 |      17 |     206.32 |     121.68 |    55.64 |     84.63 |  58.980
 1dice37Ee     128      0.00195 |     6231 |     14 (0.00226) |   6169 |      48 |    1232.64 |    1143.24 |    40.25 |     89.39 |  92.747
 1dice3jkp     256      0.00391 |     4554 |     21 (0.00462) |   4520 |      13 |     489.79 |     332.02 |    13.11 |    157.77 |  67.788
 1dice4J1m     512      0.00781 |     6743 |     42 (0.00623) |   6696 |       5 |    1451.62 |     580.08 |     9.35 |    871.54 |  39.961
 1dice5wwE    1000      0.01526 |     9926 |    145 (0.01461) |   9779 |       2 |    2009.40 |    1586.97 |     1.80 |    422.43 |  78.977
 1dice61SN    1500      0.02289 |     6888 |    160 (0.02325) |   6722 |       6 |    2900.58 |    3248.48 |    15.00 |   -347.89 | 111.994
 1dice6DPt    2000      0.03052 |     7484 |    240 (0.03208) |   7241 |       3 |    3198.76 |    2892.58 |     9.24 |    306.18 |  90.428
 1dice6gJg    3000      0.04578 |     6834 |    343 (0.05024) |   6484 |       7 |    4753.10 |    6300.69 |    24.99 |  -1547.59 | 132.560
 1dice6GV5    4000      0.06104 |     7316 |    467 (0.06386) |   6846 |       3 |    2679.37 |    2428.34 |    31.20 |    251.03 |  90.631
 1dice6wBx    6000      0.09155 |    14047 |   1311 (0.09336) |  12731 |       5 |    8490.42 |    8643.85 |     7.01 |   -153.43 | 101.807
 1dice6YgE    8000      0.12207 |    26734 |   3323 (0.12430) |  23411 |       0 |    5928.24 |    5217.30 |     0.00 |    710.93 |  88.008
 1dice7EYz   12000      0.18311 |    16004 |   3032 (0.18951) |  12967 |       5 |    6614.67 |    6748.01 |    14.50 |   -133.34 | 102.016
 1dice7fUk   16000      0.24414 |    41485 |  10041 (0.24208) |  31437 |       7 |   12997.67 |   12587.37 |    97.79 |    410.29 |  96.843
 1dice7W2A   24000      0.36621 |    30756 |  11384 (0.37052) |  19340 |      32 |   13072.95 |   13065.38 |   212.63 |      7.56 |  99.942
 1dice8EMZ   32000      0.48828 |   293350 | 142930 (0.48744) | 150293 |     127 |   89781.53 |   90624.37 |  2173.21 |   -842.83 | 100.939
 1dice97EC   32768      0.50000 |   122830 |  61255 (0.49900) |  61501 |      74 |   43565.84 |   42262.09 |   789.20 |   1303.74 |  97.007
 1dice9wcM   48000      0.73242 |    84466 |  62126 (0.73590) |  22296 |      44 |   52384.72 |   50767.66 |   467.98 |   1617.06 |  96.913
 1dice9wVt   64000      0.97656 |     5763 |   5514 (0.97835) |    122 |     127 |    5006.80 |    4813.97 |   239.20 |    192.82 |  96.149
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   714100 | 302356           | 410770 |     974 |  256969.36 |  253464.60 |  4236.52 |   3504.75 |  98.636
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3504.75907441 BTC (1.364%)
Cumulative Fees Paid:         359.47685000 BTC
SD Profit after fees:        3145.28222441 BTC (1.224%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2151576  :  SatoshiDice Tx: 1321428  (61.4%)
Blockchain MB:  909.7  :  SatoshiDice Tx: 541.6  (59.5%)

[dooglus]

Despite increasing the maximum bets far in excess of safe levels, SatoshiDice appears to be prospering:

Results: 2012-Jul-18 03:50pm (up to block 189712)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |     9763 |      0 (0.00000) |   9495 |     268 |      45.20 |       0.01 |    16.20 |     45.18 |   0.043
 1dice1Qf4       2      0.00003 |      978 |      0 (0.00000) |    909 |      69 |       7.76 |       0.00 |     5.38 |      7.76 |   0.023
 1dice2pxm       4      0.00006 |     1489 |      0 (0.00000) |   1457 |      32 |      13.08 |       0.00 |     1.22 |     13.08 |   0.043
 1dice2vQo       8      0.00012 |     1274 |      0 (0.00000) |   1235 |      39 |      17.66 |       0.00 |     3.65 |     17.66 |   0.027
 1dice2WmR      16      0.00024 |     1465 |      0 (0.00000) |   1435 |      30 |      22.83 |       0.01 |     6.60 |     22.82 |   0.074
 1dice2xkj      32      0.00049 |     3384 |      1 (0.00030) |   3372 |      11 |     105.80 |     100.40 |     1.29 |      5.40 |  94.892
 1dice2zdo      64      0.00098 |     5157 |      7 (0.00136) |   5133 |      17 |     206.72 |     121.68 |    55.64 |     85.03 |  58.865
 1dice37Ee     128      0.00195 |     6237 |     14 (0.00226) |   6175 |      48 |    1232.90 |    1143.24 |    40.25 |     89.65 |  92.728
 1dice3jkp     256      0.00391 |     4557 |     21 (0.00462) |   4523 |      13 |     489.86 |     332.02 |    13.11 |    157.83 |  67.780
 1dice4J1m     512      0.00781 |     6748 |     42 (0.00623) |   6701 |       5 |    1454.17 |     580.09 |     9.35 |    874.08 |  39.891
 1dice5wwE    1000      0.01526 |    10304 |    150 (0.01456) |  10152 |       2 |    2019.40 |    1587.57 |     1.80 |    431.82 |  78.616
 1dice61SN    1500      0.02289 |     6894 |    160 (0.02323) |   6728 |       6 |    2902.34 |    3248.48 |    15.00 |   -346.13 | 111.926
 1dice6DPt    2000      0.03052 |     7490 |    241 (0.03219) |   7246 |       3 |    3198.97 |    2894.17 |     9.24 |    304.79 |  90.472
 1dice6gJg    3000      0.04578 |     6874 |    345 (0.05024) |   6522 |       7 |    4764.14 |    6307.82 |    24.99 |  -1543.67 | 132.402
 1dice6GV5    4000      0.06104 |     7351 |    471 (0.06410) |   6877 |       3 |    2692.04 |    2457.87 |    31.20 |    234.16 |  91.301
 1dice6wBx    6000      0.09155 |    14075 |   1314 (0.09339) |  12756 |       5 |    8496.21 |    8654.54 |     7.01 |   -158.33 | 101.864
 1dice6YgE    8000      0.12207 |    27804 |   3461 (0.12448) |  24343 |       0 |    6016.42 |    5302.27 |     0.00 |    714.14 |  88.130
 1dice7EYz   12000      0.18311 |    16079 |   3045 (0.18944) |  13029 |       5 |    6635.17 |    6767.86 |    14.50 |   -132.69 | 102.000
 1dice7fUk   16000      0.24414 |    41919 |  10156 (0.24232) |  31756 |       7 |   13114.08 |   12701.42 |    97.79 |    412.66 |  96.853
 1dice7W2A   24000      0.36621 |    31314 |  11597 (0.37072) |  19685 |      32 |   13187.71 |   13193.53 |   212.63 |     -5.82 | 100.044
 1dice8EMZ   32000      0.48828 |   296595 | 144484 (0.48735) | 151984 |     127 |   91139.54 |   92043.78 |  2173.21 |   -904.23 | 100.992
 1dice97EC   32768      0.50000 |   125771 |  62715 (0.49894) |  62981 |      75 |   44818.55 |   43094.93 |   789.20 |   1723.61 |  96.154
 1dice9wcM   48000      0.73242 |    86671 |  63756 (0.73598) |  22871 |      44 |   58530.48 |   57085.16 |   467.98 |   1445.32 |  97.531
 1dice9wVt   64000      0.97656 |     5773 |   5524 (0.97839) |    122 |     127 |    5010.39 |    4817.57 |   239.20 |    192.81 |  96.152
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   725966 | 307504           | 417487 |     975 |  266121.55 |  262434.54 |  4236.52 |   3687.00 |  98.615
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3687.00665526 BTC (1.385%)
Cumulative Fees Paid:         365.44077500 BTC
SD Profit after fees:        3321.56588026 BTC (1.248%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2189111  :  SatoshiDice Tx: 1342999  (61.3%)
Blockchain MB:  926.2  :  SatoshiDice Tx: 550.7  (59.5%)

[dooglus]

3 new games have been introduced: lessthan 52k, 56k, and 60k:

Results: 2012-Jul-19 04:14pm (up to block 189848)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    10095 |      0 (0.00000) |   9826 |     269 |      49.71 |       0.01 |    16.20 |     49.70 |   0.039
 1dice1Qf4       2      0.00003 |      984 |      0 (0.00000) |    915 |      69 |       7.78 |       0.00 |     5.38 |      7.78 |   0.023
 1dice2pxm       4      0.00006 |     1493 |      0 (0.00000) |   1461 |      32 |      13.09 |       0.00 |     1.22 |     13.08 |   0.043
 1dice2vQo       8      0.00012 |     1279 |      0 (0.00000) |   1240 |      39 |      17.68 |       0.00 |     3.65 |     17.67 |   0.027
 1dice2WmR      16      0.00024 |     1470 |      0 (0.00000) |   1440 |      30 |      22.89 |       0.01 |     6.60 |     22.87 |   0.074
 1dice2xkj      32      0.00049 |     3389 |      1 (0.00030) |   3377 |      11 |     105.91 |     100.40 |     1.29 |      5.50 |  94.798
 1dice2zdo      64      0.00098 |     5162 |      7 (0.00136) |   5138 |      17 |     206.73 |     121.68 |    55.64 |     85.04 |  58.863
 1dice37Ee     128      0.00195 |     6247 |     14 (0.00226) |   6185 |      48 |    1233.06 |    1143.24 |    40.25 |     89.81 |  92.716
 1dice3jkp     256      0.00391 |     4562 |     21 (0.00462) |   4528 |      13 |     490.07 |     332.02 |    13.11 |    158.04 |  67.750
 1dice4J1m     512      0.00781 |     6775 |     42 (0.00620) |   6728 |       5 |    1456.03 |     580.09 |     9.35 |    875.93 |  39.841
 1dice5wwE    1000      0.01526 |    10486 |    150 (0.01431) |  10334 |       2 |    2051.62 |    1587.69 |     1.80 |    463.92 |  77.388
 1dice61SN    1500      0.02289 |     6898 |    160 (0.02322) |   6732 |       6 |    2902.66 |    3248.48 |    15.00 |   -345.82 | 111.914
 1dice6DPt    2000      0.03052 |     7521 |    243 (0.03232) |   7275 |       3 |    3202.11 |    2904.42 |     9.24 |    297.69 |  90.703
 1dice6gJg    3000      0.04578 |     6914 |    348 (0.05038) |   6559 |       7 |    4767.11 |    6317.21 |    24.99 |  -1550.09 | 132.516
 1dice6GV5    4000      0.06104 |     7399 |    473 (0.06395) |   6923 |       3 |    2699.06 |    2458.54 |    31.20 |    240.52 |  91.089
 1dice6wBx    6000      0.09155 |    14175 |   1327 (0.09365) |  12843 |       5 |    8502.09 |    8668.21 |     7.01 |   -166.12 | 101.954
 1dice6YgE    8000      0.12207 |    28522 |   3556 (0.12471) |  24959 |       7 |    6058.32 |    5341.87 |     0.00 |    716.44 |  88.174
 1dice7EYz   12000      0.18311 |    16117 |   3051 (0.18936) |  13061 |       5 |    6638.82 |    6772.67 |    14.50 |   -133.85 | 102.016
 1dice7fUk   16000      0.24414 |    42142 |  10219 (0.24253) |  31916 |       7 |   13161.09 |   12747.59 |    97.79 |    413.49 |  96.858
 1dice7W2A   24000      0.36621 |    31588 |  11696 (0.37064) |  19860 |      32 |   13225.56 |   13222.07 |   212.63 |      3.49 |  99.974
 1dice8EMZ   32000      0.48828 |   301258 | 146729 (0.48726) | 154400 |     129 |   92456.79 |   93222.19 |  2173.21 |   -765.39 | 100.828
 1dice97EC   32768      0.50000 |   126199 |  62916 (0.49885) |  63207 |      76 |   45118.62 |   43391.34 |   789.20 |   1727.28 |  96.172
 1dice9wcM   48000      0.73242 |    88026 |  64754 (0.73600) |  23227 |      45 |   62057.75 |   60478.44 |   467.98 |   1579.30 |  97.455
 1dicec9k7   52000      0.79346 |       23 |     18 (0.78261) |      5 |       0 |      10.68 |      11.64 |     0.00 |     -0.96 | 109.002
 1dicegEAr   56000      0.85449 |       11 |      7 (0.63636) |      4 |       0 |       0.32 |       0.27 |     0.00 |      0.04 |  85.085
 1diceDCd2   60000      0.91553 |        4 |      3 (0.75000) |      1 |       0 |       2.55 |       2.71 |     0.00 |     -0.16 | 106.514
 1dice9wVt   64000      0.97656 |     5794 |   5545 (0.97847) |    122 |     127 |    5012.36 |    4819.54 |   239.20 |    192.81 |  96.153
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   734533 | 311280           | 422266 |     987 |  271470.56 |  267472.47 |  4236.53 |   3998.08 |  98.527
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3998.08798832 BTC (1.473%)
Cumulative Fees Paid:         369.74470000 BTC
SD Profit after fees:        3628.34328832 BTC (1.337%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2218777  :  SatoshiDice Tx: 1359223  (61.3%)
Blockchain MB:  939.4  :  SatoshiDice Tx: 557.9  (59.4%)

[organofcorti]

The house edge is determined by the payout when you win.  The chance of winning is always the same - it picks a number between 0 and 65535 and you win if the number is less than 8000.  Your calculation is off a little because 1 - 8000/65536 isn't 7/8, it's a little more than that.

I make it a one-in-17399 chance of losing all 75 bets, so still really unlucky.  But then again 1-in-17k shots do happen.  About once every 17k bets.  And there have been over 600k bets.

 >>> 1 / ((1 - 8000/65536.0) ** 75)
  17398.822919111808

Quick comment: It's even unluckier than that, since  it's 1 chance in 17399 of a run of 75 or less bets in a row not winning - so that's going to have required 17399 * 75 =   1304925 or more bets to make it likely to have occurred once.

Enjoying you analysis so far - thanks for your work.

[dooglus]

Quick comment: It's even unluckier than that, since  it's 1 chance in 17399 of a run of 75 or less bets in a row not winning - so that's going to have required 17399 * 75 =   1304925 or more bets to make it likely to have occurred once.

Enjoying you analysis so far - thanks for your work.

I'm not sure, but I don't think you should multiply by 75.  Maybe add (75-1).

Consider flipping a fair coin.  How many times do you need to flip before it is 'likely' to get 10 heads in a row?

It's a 1-in-1024 chance, so I would say you need to flip 1024 + (10-1) = 1033 coins to have a reasonable chance of 310 heads in a row.

By your (implied) logic would you say you need 1024*10 = 10240 flips to make it likely?

I just ran a simulation.  Flipping coins until getting 10 in a row, and counting how many flips it took.  I tried it 100,000 times.  116 (0.12%) of those times the first 10 coins were all heads.  To have a 1% chance of getting 10 heads in a row I had to flip 29 times.  For 10% I had to flip 221 coins, etc.  Flipping 1033 coins gives a 39.71% chance of getting 10 heads in a row.

But flipping 10239 coins gives a huge 99.34% chance.

I have a feeling this is a binomial distribution thing, so I should be able to work out the maths rather than doing it by simulation.  But I think that after 17399 * 75 plays you're almost guaranteed to have a losing streak of 75 or more.

Oh, and the analysis I'm posting is based on work done by etotheipi - the OP in this thread.  But I'm glad you appreciate it.

[organofcorti]

I think you might be right - serves me right for posting while trying to leave early for lunch. I'll go over it later and see if I come to the same results as you.

[dooglus]

Here's the simulation code:

#!/usr/bin/python

import random

trials = 100000
target = 10
max = 2**target * target

count =

• * max

for trial in range(trials):
    in_a_row = 0
    for flip in range(max):
        if random.random() < 0.5:
            in_a_row += 1
            if in_a_row == target:
                count[flip] += 1
                break
        else:
            in_a_row = 0

cumulative = 0
for flip in range(max):
    cumulative += count[flip]
    if count[flip]:
        print("%4d on flip %d; cumulative %6.2f%%" %
              (count[flip],
               flip+1,
               100.00 * cumulative / trials))

[organofcorti]

OK, here's how I see it:

The number of times any one player must play before he wins at the 8000/65536 game is a shifted geometrically distributed random variable, p = 8000/65536. Let the term "round" indicate a series of plays ending in a win.

This means the probability of 75 or more plays before a win is given by the upper tail geometric distribution CDF:

Code:

(1-8000/65536)^75 =  5.747515e-05

So for every round played, there is a 5.747515e-05 chance the round will end after 74 plays. So, on average, we expect one in 17398.82 rounds to last 75 plays or more.

The mean number of plays per round is:

Code:

1/p = 8.192

So, converting rounds to plays, we expect a run of 75 or more losses in a row to occur once in 17398.82 * 8.192 = 142531.1 plays.

So I was out by a factor of almost 10.


For you (fair) coin flip example, calling a series of plays ending in a head a "round", 10 heads in a row has a probability of occurring determined by the probability density function, p*(1-p)^(k-1).

In this case, k = 10 and p = 0.5, so the probability of winning exactly 10 heads in a row for a given round is 0.5*(1-0.5)^9 = 1024

There is a mean of 2 flips per game, so we'd expect exactly 10 heads in a row once every 2048 flips.

Here is my code for a coin flip simulation, written in R:

Code:

> rg   <- rgeom(1e06, 1/2) + 1
> length(which(rg==10))/sum(rg)
[1] 0.0004780581

1/0.0004780581 = 2091.796, quite close to the exact 2048.


Edit: If you want to know the probability of exactly 75 losses in a row occurring, use the geometric distribution probability density function: p*(1-p)^(k-1) =  8000/65536*(1- 8000/65536)^(75-1) = 7.99154e-06 or once in 125132.3 rounds. Since one round has a mean number of plays of 8.192, the expected number of plays before a round of exactly 75 losses before one win is 125132.3 * 8.192 = 1025084 plays.

Some more R code for you to try, simulating a game that ends a series of losses with a win, the probability of a win being 8000/65536:

Code:

> rg   <- rgeom(1e07, 8000/65536) + 1
> 1/(length(which(rg==75))/sum(rg))
[1] 1092795

Quite close. The lower p is the greater the variance so you might want to bump the first term in rgeom() up to 1e10 if you machines faster than my work computer.

[dooglus]

There is a mean of 2 flips per game, so we'd expect exactly 10 heads in a row once every 2048 flips.

Is that 'exactly' where we are differing?  The guy didn't have exactly 75 losses and then a win.  He had 75 losses and quit playing.  Who knows how many losses he would have had if he had kept playing.

I think we expect at least 10 heads in a row once every 1024 flips.

(exactly 11 heads is half as likely as exactly 10, exactly 12 is half as likely again,etc.  1/2 + 1/4 + ... = 1.  ie. the chance of seeing exactly 10 is the same as the chance of seeing 11 or more.  I see exactly 10 once every 2048 flips, I see 11 or more once every 2048 flips too.  Combining these 2, I see 10 or more every 1024 flips.

[organofcorti]

There is a mean of 2 flips per game, so we'd expect exactly 10 heads in a row once every 2048 flips.

Is that 'exactly' where we are differing?  The guy didn't have exactly 75 losses and then a win.  He had 75 losses and quit playing.  Who knows how many losses he would have had if he had kept playing.

Then he hasn't finished playing, in terms of the definition of "round" which I gave above. This is equivalent to "75 losses or greater" the probability which is as I described:

This means the probability of 75 or more plays before a win is given by the upper tail geometric distribution CDF

except I was wrong - the upper tail CDF is (1-1/p)^(k-1). Ooops! Still, the basic idea remains the same - multiply by the number of expected rounds per game to get the number of plays before 75 losses on a row or more occurs.

I think we expect at least 10 heads in a row once every 1024 flips.

(exactly 11 heads is half as likely as exactly 10, exactly 12 is half as likely again,etc.  1/2 + 1/4 + ... = 1.  ie. the chance of seeing exactly 10 is the same as the chance of seeing 11 or more.  I see exactly 10 once every 2048 flips, I see 11 or more once every 2048 flips too.  Combining these 2, I see 10 or more every 1024 flips.

Sure. As per the first part of my post, you can find the probability of 10 or more heads by using the upper tail CDF: (1-p)^(k-1). In this case, it's (1-1/2)^9 = 1/512. This probability means that for a round defined as a series of flips ending in a tail, flipping 10 or more heads in a row should occur once every 512 rounds, or once every 1024 plays - as you saw in your simulation.

So going back to the original question - how many times would any one players be expected to play to experience 75 losses or more in a row? The upper tail CDF is (1-8000/65536)^(75-1) = 1/15274.94  and the mean number of plays per round = 8.192, so we would expected a run of 75 losses or greater to occur once every  125132.3 plays. Different from my original answer because I had the upper tail CDF function wrong. You can check with this simulation if you like:

Code:

> rg   <- rgeom(1e07, 8000/65536) + 1
> 1/(length(which(rg>=75))/sum(rg))
[1] 129051.3

I could be wrong - write your own sim to check and see how it goes!

[dooglus]

You're right.  It seems counter-intuitive to me that flipping three heads in a row (a 1-in-8 shot) takes an expected 14 flips to happen, but it does.

Because the average length of a round which ends when we either flip a tail or get 3 heads:

T : half the time
HT : 1/4 the time
HHT : 1/8 the time
HHH : 1/8 the time

is 1*1/2 + 2*1/4 + 3*1/8 + 3*1/8 = 1.75 tosses - so the expectation is that it takes 1.75*8 = 14 flips.

Thanks for persevering with me!