Satoshi Dice -- Statistical Analysis

[organofcorti]

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.

Tricky one, that. Toss a coin one million times and see how many times you get three heads in a row, right? Not a geometrically distributed random anymore, since the round end is either throwing a tail or or getting three heads in a row.

I'm not sure of your method but you do have the correct answer. In general, the number of coin tosses it takes on average to get n heads in a row is given by:

Code:

(p^(-n) - 1)/(1 - p), where p = 0.5 and n = the number of heads in a row

  Heads in a row   Expect. no. tosses
               1                  2
               2                  6
               3                 14
               4                 30
               5                 62
               6                126
               7                254
               8                510
               9               1022
             10               2046
             11               4094
             12               8190
             13              16382
             14              32766
             15              65534
             16             131070
             17             262142
             18             524286
             19            1048574
             20            2097150

Try this against your method and see if you're right in a few more cases. I'm not saying you're wrong, just that I don't follow how you got your result.

There's a very nice derivation of this (and of a more general solution where the probability of a head or tail varies with each toss) using Markov chains here.

Thanks for persevering with me!

It was fun and I got to learn some new things. Plus you also persevered - I could have been wrong.

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[dooglus]

I'm not sure of your method but you do have the correct answer. In general, the number of coin tosses it takes on average to get n heads in a row is given by:

Code:

(p^(-n) - 1)/(1 - p), where p = 0.5 and n = the number of heads in a row

  Heads in a row   Expect. no. tosses
               1                  2
               2                  6
               3                 14
               4                 30
               5                 62
               6                126

Try this against your method and see if you're right in a few more cases. I'm not saying you're wrong, just that I don't follow how you got your result.

Here, for 6 heads in a row.  The expected length of a round is the sum of the probability of each outcome times the length of that outcome:

Outcome	p(Outcome)	len(Outcome)	p(Outcome)*len(Outcome)
T	1/2	1	1/2
HT	1/4	2	1/2
HHT	1/8	3	3/8
HHHT	1/16	4	1/4
HHHHT	1/32	5	5/32
HHHHHT	1/64	6	3/32
HHHHHH	1/64	6	3/32
			-----
			63/32

so 63/32 flips.  The expected number of rounds to get 6 heads is 64, the expected length of each round is 63/32, so the expected number of flips is 64*63/32 = 126.

It was fun and I got to learn some new things. Plus you also persevered - I could have been wrong.

... and you were, a couple of times...    But only in little ways.

This reminded me of both Penney's game and the discussion around this question, both of which you may find interesting if you didn't see them before.

[dooglus]

After all that unpleasant maths back the your regularly scheduled tables of numbers and graphs...

Results: 2012-Jul-20 11:44am (up to block 189971)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP  
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    10450 |      0 (0.00000) |  10180 |     270 |      51.83 |       0.01 |    16.30 |     51.82 |   0.037
 1dice1Qf4       2      0.00003 |      989 |      0 (0.00000) |    920 |      69 |       7.94 |       0.00 |     5.38 |      7.94 |   0.022
 1dice2pxm       4      0.00006 |     1498 |      0 (0.00000) |   1466 |      32 |      13.55 |       0.00 |     1.22 |     13.55 |   0.049
 1dice2vQo       8      0.00012 |     1284 |      0 (0.00000) |   1244 |      40 |      18.14 |       0.00 |     4.15 |     18.13 |   0.034
 1dice2WmR      16      0.00024 |     1476 |      0 (0.00000) |   1446 |      30 |      24.37 |       0.02 |     6.60 |     24.34 |   0.094
 1dice2xkj      32      0.00049 |     3417 |      1 (0.00029) |   3405 |      11 |     106.99 |     100.40 |     1.29 |      6.59 |  93.841
 1dice2zdo      64      0.00098 |     5167 |      7 (0.00136) |   5143 |      17 |     209.30 |     121.70 |    55.64 |     87.60 |  58.146
 1dice37Ee     128      0.00195 |     6257 |     14 (0.00225) |   6195 |      48 |    1233.36 |    1143.24 |    40.25 |     90.11 |  92.693
 1dice3jkp     256      0.00391 |     4609 |     21 (0.00457) |   4574 |      14 |     491.61 |     332.02 |    13.11 |    159.58 |  67.538
 1dice4J1m     512      0.00781 |     6931 |     43 (0.00621) |   6883 |       5 |    1460.76 |     580.23 |     9.35 |    880.52 |  39.721
 1dice5wwE    1000      0.01526 |    12041 |    169 (0.01404) |  11870 |       2 |    2089.30 |    1665.78 |     1.80 |    423.51 |  79.729
 1dice61SN    1500      0.02289 |     7031 |    163 (0.02320) |   6862 |       6 |    2947.56 |    3355.32 |    15.00 |   -407.76 | 113.834
 1dice6DPt    2000      0.03052 |     7634 |    244 (0.03197) |   7387 |       3 |    3242.60 |    2968.55 |     9.24 |    274.04 |  91.549
 1dice6gJg    3000      0.04578 |     7038 |    350 (0.04978) |   6681 |       7 |    4810.53 |    6361.10 |    24.99 |  -1550.57 | 132.233
 1dice6GV5    4000      0.06104 |     7602 |    487 (0.06409) |   7112 |       3 |    2775.87 |    2565.27 |    31.20 |    210.59 |  92.413
 1dice6wBx    6000      0.09155 |    14370 |   1355 (0.09433) |  13010 |       5 |    8526.55 |    8694.98 |     7.01 |   -168.42 | 101.975
 1dice6YgE    8000      0.12207 |    29138 |   3640 (0.12495) |  25491 |       7 |    6098.25 |    5379.06 |     0.00 |    719.19 |  88.207
 1dice7EYz   12000      0.18311 |    16210 |   3072 (0.18957) |  13133 |       5 |    6661.61 |    6807.11 |    14.50 |   -145.49 | 102.184
 1dice7fUk   16000      0.24414 |    42396 |  10284 (0.24261) |  32105 |       7 |   13255.13 |   12875.71 |    97.79 |    379.42 |  97.138
 1dice7W2A   24000      0.36621 |    31849 |  11782 (0.37031) |  20035 |      32 |   13330.29 |   13301.97 |   212.63 |     28.32 |  99.787
 1dice8EMZ   32000      0.48828 |   304780 | 148509 (0.48747) | 156142 |     129 |   93709.98 |   94614.77 |  2173.21 |   -904.78 | 100.966
 1dice97EC   32768      0.50000 |   127050 |  63345 (0.49888) |  63629 |      76 |   46098.86 |   44536.87 |   789.20 |   1561.98 |  96.612
 1dice9wcM   48000      0.73242 |    89590 |  65896 (0.73590) |  23649 |      45 |   65018.36 |   63475.25 |   467.98 |   1543.10 |  97.627
 1dicec9k7   52000      0.79346 |      157 |    123 (0.78344) |     34 |       0 |      61.72 |      60.32 |     0.00 |      1.39 |  97.733
 1dicegEAr   56000      0.85449 |       51 |     39 (0.76471) |     12 |       0 |     131.38 |      93.56 |     0.00 |     37.82 |  71.213
 1diceDCd2   60000      0.91553 |       22 |     17 (0.77273) |      5 |       0 |      15.59 |      15.31 |     0.00 |      0.27 |  98.235
 1dice9wVt   64000      0.97656 |     5809 |   5560 (0.97853) |    122 |     127 |    5013.48 |    4820.66 |   239.20 |    192.82 |  96.154
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   744846 | 315121           | 428735 |     990 |  277405.04 |  273869.34 |  4237.13 |   3535.69 |  98.725
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3535.69622366 BTC (1.275%)
Cumulative Fees Paid:         374.93467500 BTC
SD Profit after fees:        3160.76154866 BTC (1.139%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2249769  :  SatoshiDice Tx: 1376978  (61.2%)
Blockchain MB:  952.8  :  SatoshiDice Tx: 565.8  (59.4%)

I noticed a sharp drop of 100 BTC or so on the red line about a day ago and looked to see what happened.

This bet is responsible for the loss.  2 BTC on each of lessthan 1500, 2000, 3000, and 4000, with a lucky number of 920 winning all 4 bets.  A 223.90897599 BTC return for an 8 BTC outlay!

Bet: lessthan 3000
Bet Amount: 2.00000000
Payment: 42.65159066

Bet: lessthan 2000
Bet Amount: 2.00000000
Payment: 63.97263600

Bet: lessthan 1500
Bet Amount: 2.00000000
Payment: 85.29368133

Bet: lessthan 4000
Bet Amount: 2.00000000
Payment: 31.99106800

[organofcorti]

Here, for 6 heads in a row.  The expected length of a round is the sum of the probability of each outcome times the length of that outcome:

.......

so 63/32 flips.  The expected number of rounds to get 6 heads is 64, the expected length of each round is 63/32, so the expected number of flips is 64*63/32 = 126.

I follow now. Nice clear way to approach the problem when you put it like that.

This reminded me of both Penney's game

I hadn't been aware of that, it's an interesting problem. Thanks!

Btw, what are you using for charts?

[dooglus]

This reminded me of both Penney's game

I hadn't been aware of that, it's an interesting problem. Thanks!

Btw, what are you using for charts?

I was made aware of it by one of Derren Brown's TV shows - possibly this one.  I'd check but I think the storm outside is making my Internet connection too flaky to do anything much at the moment.

The charts are from gnuplot using data generated by a modified version of the extras/sample_armory_code.py script in the armory git repository.

[dooglus]

Results: 2012-Jul-21 11:24am (up to block 190118)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    10512 |      0 (0.00000) |  10242 |     270 |      52.82 |       0.01 |    16.30 |     52.80 |   0.036
 1dice1Qf4       2      0.00003 |      990 |      0 (0.00000) |    921 |      69 |       8.04 |       0.00 |     5.38 |      8.04 |   0.022
 1dice2pxm       4      0.00006 |     1498 |      0 (0.00000) |   1466 |      32 |      13.55 |       0.00 |     1.22 |     13.55 |   0.049
 1dice2vQo       8      0.00012 |     1284 |      0 (0.00000) |   1244 |      40 |      18.14 |       0.00 |     4.15 |     18.13 |   0.034
 1dice2WmR      16      0.00024 |     1477 |      0 (0.00000) |   1447 |      30 |      24.42 |       0.02 |     6.60 |     24.39 |   0.094
 1dice2xkj      32      0.00049 |     3417 |      1 (0.00029) |   3405 |      11 |     106.99 |     100.40 |     1.29 |      6.59 |  93.841
 1dice2zdo      64      0.00098 |     5167 |      7 (0.00136) |   5143 |      17 |     209.30 |     121.70 |    55.64 |     87.60 |  58.146
 1dice37Ee     128      0.00195 |     6260 |     14 (0.00225) |   6198 |      48 |    1233.38 |    1143.24 |    40.25 |     90.13 |  92.692
 1dice3jkp     256      0.00391 |     4638 |     21 (0.00454) |   4603 |      14 |     494.40 |     332.03 |    13.11 |    162.37 |  67.159
 1dice4J1m     512      0.00781 |     7067 |     44 (0.00623) |   7018 |       5 |    1475.25 |     592.75 |     9.35 |    882.49 |  40.180
 1dice5wwE    1000      0.01526 |    13311 |    186 (0.01398) |  13123 |       2 |    2140.42 |    1683.04 |     1.80 |    457.37 |  78.631
 1dice61SN    1500      0.02289 |     7304 |    171 (0.02343) |   7127 |       6 |    2974.84 |    3383.09 |    15.00 |   -408.24 | 113.723
 1dice6DPt    2000      0.03052 |     7957 |    255 (0.03206) |   7699 |       3 |    3302.47 |    3015.76 |     9.24 |    286.71 |  91.318
 1dice6gJg    3000      0.04578 |     7341 |    364 (0.04963) |   6970 |       7 |    4856.92 |    6429.68 |    24.99 |  -1572.76 | 132.382
 1dice6GV5    4000      0.06104 |     7889 |    503 (0.06378) |   7383 |       3 |    2888.61 |    2689.64 |    31.20 |    198.97 |  93.112
 1dice6wBx    6000      0.09155 |    14890 |   1406 (0.09446) |  13479 |       5 |    8604.42 |    8781.98 |     7.01 |   -177.55 | 102.064
 1dice6YgE    8000      0.12207 |    30147 |   3774 (0.12522) |  26366 |       7 |    6206.82 |    5497.96 |     0.00 |    708.86 |  88.579
 1dice7EYz   12000      0.18311 |    16380 |   3106 (0.18968) |  13269 |       5 |    6750.36 |    6908.21 |    14.50 |   -157.85 | 102.338
 1dice7fUk   16000      0.24414 |    43006 |  10434 (0.24266) |  32565 |       7 |   13545.31 |   13113.43 |    97.79 |    431.88 |  96.812
 1dice7W2A   24000      0.36621 |    32168 |  11900 (0.37030) |  20236 |      32 |   13667.82 |   13608.03 |   212.63 |     59.79 |  99.563
 1dice8EMZ   32000      0.48828 |   308319 | 150251 (0.48753) | 157939 |     129 |   95502.88 |   96267.43 |  2173.21 |   -764.55 | 100.801
 1dice97EC   32768      0.50000 |   128228 |  63938 (0.49893) |  64213 |      77 |   48099.37 |   46629.50 |   789.20 |   1469.86 |  96.944
 1dice9wcM   48000      0.73242 |    90649 |  66699 (0.73616) |  23905 |      45 |   67451.56 |   65853.71 |   467.98 |   1597.84 |  97.631
 1dicec9k7   52000      0.79346 |      350 |    272 (0.77714) |     78 |       0 |     184.19 |     179.40 |     0.00 |      4.78 |  97.400
 1dicegEAr   56000      0.85449 |       99 |     76 (0.76768) |     23 |       0 |     196.36 |     148.35 |     0.00 |     48.00 |  75.552
 1diceDCd2   60000      0.91553 |       38 |     32 (0.84211) |      6 |       0 |      22.33 |      22.17 |     0.00 |      0.15 |  99.302
 1dice9wVt   64000      0.97656 |     5823 |   5572 (0.97857) |    122 |     129 |    5015.57 |    4822.75 |   239.20 |    192.81 |  96.156
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   756209 | 319026           | 436190 |     993 |  285046.66 |  281324.42 |  4237.13 |   3722.24 |  98.694
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3722.24804889 BTC (1.306%)
Cumulative Fees Paid:         380.64905000 BTC
SD Profit after fees:        3341.59899889 BTC (1.172%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2286024  :  SatoshiDice Tx: 1396679  (61.1%)
Blockchain MB:  967.7  :  SatoshiDice Tx: 574.1  (59.3%)

[dooglus]

Results: 2012-Jul-22 04:03pm (up to block 190291)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    10612 |      0 (0.00000) |  10333 |     279 |      53.52 |       0.01 |    17.86 |     53.51 |   0.036
 1dice1Qf4       2      0.00003 |      993 |      0 (0.00000) |    923 |      70 |       8.24 |       0.00 |     5.58 |      8.24 |   0.021
 1dice2pxm       4      0.00006 |     1499 |      0 (0.00000) |   1467 |      32 |      13.75 |       0.00 |     1.22 |     13.75 |   0.052
 1dice2vQo       8      0.00012 |     1285 |      0 (0.00000) |   1245 |      40 |      18.54 |       0.00 |     4.15 |     18.53 |   0.042
 1dice2WmR      16      0.00024 |     1480 |      0 (0.00000) |   1450 |      30 |      24.45 |       0.02 |     6.60 |     24.42 |   0.094
 1dice2xkj      32      0.00049 |     3442 |      1 (0.00029) |   3430 |      11 |     109.99 |     100.41 |     1.29 |      9.58 |  91.289
 1dice2zdo      64      0.00098 |     5178 |      7 (0.00136) |   5154 |      17 |     210.92 |     121.70 |    55.64 |     89.22 |  57.700
 1dice37Ee     128      0.00195 |     6262 |     14 (0.00225) |   6200 |      48 |    1233.88 |    1143.24 |    40.25 |     90.63 |  92.654
 1dice3jkp     256      0.00391 |     4662 |     21 (0.00452) |   4627 |      14 |     495.11 |     332.03 |    13.11 |    163.07 |  67.063
 1dice4J1m     512      0.00781 |     7173 |     46 (0.00642) |   7122 |       5 |    1514.92 |     717.83 |     9.35 |    797.08 |  47.384
 1dice5wwE    1000      0.01526 |    13523 |    193 (0.01427) |  13328 |       2 |    2208.67 |    1888.68 |     1.80 |    319.98 |  85.512
 1dice61SN    1500      0.02289 |     7503 |    175 (0.02334) |   7322 |       6 |    3046.90 |    3468.65 |    15.00 |   -421.74 | 113.842
 1dice6DPt    2000      0.03052 |     8696 |    279 (0.03209) |   8414 |       3 |    3386.76 |    3114.94 |     9.24 |    271.81 |  91.974
 1dice6gJg    3000      0.04578 |     7481 |    367 (0.04910) |   7107 |       7 |    4904.68 |    6452.27 |    24.99 |  -1547.58 | 131.553
 1dice6GV5    4000      0.06104 |     8246 |    522 (0.06333) |   7721 |       3 |    2999.67 |    2777.53 |    31.20 |    222.14 |  92.594
 1dice6wBx    6000      0.09155 |    15166 |   1426 (0.09406) |  13735 |       5 |    8653.84 |    8830.60 |     7.01 |   -176.76 | 102.043
 1dice6YgE    8000      0.12207 |    30631 |   3823 (0.12484) |  26801 |       7 |    6255.83 |    5535.22 |     0.00 |    720.60 |  88.481
 1dice7EYz   12000      0.18311 |    16525 |   3136 (0.18983) |  13384 |       5 |    6785.06 |    6957.16 |    14.50 |   -172.09 | 102.536
 1dice7fUk   16000      0.24414 |    43181 |  10481 (0.24276) |  32693 |       7 |   13578.67 |   13151.00 |    97.79 |    427.66 |  96.850
 1dice7W2A   24000      0.36621 |    32517 |  12037 (0.37054) |  20448 |      32 |   13709.59 |   13652.30 |   212.63 |     57.28 |  99.582
 1dice8EMZ   32000      0.48828 |   311497 | 151841 (0.48766) | 159525 |     131 |   96191.56 |   97008.25 |  2173.21 |   -816.69 | 100.849
 1dice97EC   32768      0.50000 |   129506 |  64573 (0.49891) |  64854 |      79 |   48455.42 |   46905.01 |   789.20 |   1550.41 |  96.800
 1dice9wcM   48000      0.73242 |    92779 |  68268 (0.73621) |  24461 |      50 |   74075.16 |   72580.05 |   467.98 |   1495.11 |  97.982
 1dicec9k7   52000      0.79346 |      712 |    570 (0.80056) |    142 |       0 |    1184.15 |    1200.70 |     0.00 |    -16.55 | 101.398
 1dicegEAr   56000      0.85449 |      336 |    272 (0.80952) |     64 |       0 |     234.41 |     184.05 |     0.00 |     50.35 |  78.518
 1diceDCd2   60000      0.91553 |       58 |     51 (0.87931) |      7 |       0 |      37.58 |      38.23 |     0.00 |     -0.65 | 101.732
 1dice9wVt   64000      0.97656 |     5850 |   5599 (0.97868) |    122 |     129 |    5021.19 |    4828.39 |   239.20 |    192.80 |  96.160
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   766793 | 323702           | 442079 |    1012 |  294412.58 |  290988.41 |  4238.89 |   3424.17 |  98.837
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3424.17659660 BTC (1.163%)
Cumulative Fees Paid:         385.98247500 BTC
SD Profit after fees:        3038.19412160 BTC (1.032%)
----
Since Satoshi Dice started, there have been:>
Blockchain Tx:  2325411  :  SatoshiDice Tx: 1416455  (60.9%)
Blockchain MB:  983.2  :  SatoshiDice Tx: 582.3  (59.2%)

[organofcorti]

I thought this might be of interest. The last column of the table shows how probable the luck for a given game is. As a simple indicator, consider game with a CDF over 0.95 to be very lucky and any game with a CDF under 0.05 to be very unlucky.

As a measure is's quite useful since the CDF results tends to be counter-intuitive as they are affected by observation size as well as the outcome. For example, the 48000 game has been very lucky so far, even though the fraction of wins (0.73581) is not much larger than the probability (0.73242). The 128 game seems very lucky (0.00224 compared with 0.00195) but the CDF shows us this is not significantly lucky (0.75278). The 32 game seems unlucky (0.00029 cw 0.00049) but isn't unusual  (0.49737).

[dooglus]

Interesting.

How are you calculating the CDF?

[organofcorti]

I used the pbinom function in R to calculate a binomial CDF, instead of messing with beta functions. You could also use the Gnu Scientific Library, specifically gsl_cdf_binomial_P where the three variables used in the function are the number of wins (q in R, k in gsl), the probability (p in both R and gsl), and the number of trials (size in R, n in gsl).

For my variables I used your data with the integer value in the "Win" column as q, the "Should win" column as p, and the "#Bets" column as size. If you're using R, also use the lower.tail = TRUE arg.

[dooglus]

Results: 2012-Jul-24 01:47am (up to block 190505)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    10762 |      0 (0.00000) |  10482 |     280 |      57.71 |       0.01 |    17.88 |     57.69 |   0.033
 1dice1Qf4       2      0.00003 |      999 |      0 (0.00000) |    929 |      70 |       8.54 |       0.00 |     5.58 |      8.54 |   0.021
 1dice2pxm       4      0.00006 |     1506 |      0 (0.00000) |   1474 |      32 |      13.88 |       0.00 |     1.22 |     13.87 |   0.051
 1dice2vQo       8      0.00012 |     1289 |      0 (0.00000) |   1249 |      40 |      18.64 |       0.00 |     4.15 |     18.64 |   0.042
 1dice2WmR      16      0.00024 |     1514 |      0 (0.00000) |   1484 |      30 |      25.62 |       0.02 |     6.60 |     25.59 |   0.103
 1dice2xkj      32      0.00049 |     3455 |      1 (0.00029) |   3443 |      11 |     111.43 |     100.41 |     1.29 |     11.02 |  90.111
 1dice2zdo      64      0.00098 |     5210 |      7 (0.00135) |   5186 |      17 |     212.08 |     121.70 |    55.64 |     90.37 |  57.385
 1dice37Ee     128      0.00195 |     6325 |     15 (0.00239) |   6262 |      48 |    1234.78 |    1148.24 |    40.25 |     86.53 |  92.992
 1dice3jkp     256      0.00391 |     4874 |     22 (0.00453) |   4838 |      14 |     516.89 |     334.58 |    13.11 |    182.30 |  64.730
 1dice4J1m     512      0.00781 |     7647 |     49 (0.00641) |   7593 |       5 |    1560.53 |     746.69 |     9.35 |    813.84 |  47.848
 1dice5wwE    1000      0.01526 |    14013 |    205 (0.01463) |  13806 |       2 |    2299.43 |    1970.99 |     1.80 |    328.44 |  85.716
 1dice61SN    1500      0.02289 |     7801 |    183 (0.02348) |   7612 |       6 |    3073.99 |    3498.29 |    15.00 |   -424.30 | 113.803
 1dice6DPt    2000      0.03052 |     9469 |    298 (0.03148) |   9168 |       3 |    3437.89 |    3149.86 |     9.24 |    288.03 |  91.622
 1dice6gJg    3000      0.04578 |     7691 |    378 (0.04919) |   7306 |       7 |    4950.12 |    6483.39 |    24.99 |  -1533.26 | 130.974
 1dice6GV5    4000      0.06104 |     8533 |    543 (0.06366) |   7987 |       3 |    3045.70 |    2822.91 |    31.20 |    222.78 |  92.685
 1dice6wBx    6000      0.09155 |    15779 |   1491 (0.09453) |  14282 |       6 |    8760.70 |    8957.31 |     7.01 |   -196.61 | 102.244
 1dice6YgE    8000      0.12207 |    31680 |   3944 (0.12452) |  27729 |       7 |    6387.25 |    5654.73 |     0.00 |    732.51 |  88.532
 1dice7EYz   12000      0.18311 |    16580 |   3146 (0.18980) |  13429 |       5 |    6845.20 |    6994.97 |    14.50 |   -149.76 | 102.188
 1dice7fUk   16000      0.24414 |    44182 |  10721 (0.24269) |  33454 |       7 |   13861.13 |   13682.86 |    97.79 |    178.27 |  98.714
 1dice7W2A   24000      0.36621 |    32836 |  12157 (0.37060) |  20647 |      32 |   13747.50 |   13676.13 |   212.63 |     71.36 |  99.481
 1dice8EMZ   32000      0.48828 |   315841 | 153941 (0.48760) | 161769 |     131 |   98000.04 |   98557.45 |  2173.21 |   -557.40 | 100.569
 1dice97EC   32768      0.50000 |   130873 |  65255 (0.49891) |  65539 |      79 |   48823.95 |   47197.33 |   789.20 |   1626.61 |  96.668
 1dice9wcM   48000      0.73242 |    93996 |  69156 (0.73613) |  24790 |      50 |   76324.51 |   74924.62 |   467.98 |   1399.89 |  98.166
 1dicec9k7   52000      0.79346 |      875 |    700 (0.80000) |    175 |       0 |    1335.75 |    1344.51 |     0.00 |     -8.76 | 100.656
 1dicegEAr   56000      0.85449 |      689 |    569 (0.82583) |    120 |       0 |     407.99 |     343.46 |     0.00 |     64.52 |  84.184
 1diceDCd2   60000      0.91553 |      140 |    125 (0.91241) |     12 |       3 |      48.09 |      48.61 |     0.00 |     -0.52 | 101.093
 1dice9wVt   64000      0.97656 |     5906 |   5653 (0.97854) |    124 |     129 |    5025.32 |    4832.49 |   239.20 |    192.83 |  96.163
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   780465 | 328559           | 450889 |    1017 |  300134.79 |  296591.69 |  4238.91 |   3543.10 |  98.819
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       3543.10320692 BTC (1.181%)
Cumulative Fees Paid:         392.85462500 BTC
SD Profit after fees:        3150.24858192 BTC (1.050%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2373598  :  SatoshiDice Tx: 1441132  (60.7%)
Blockchain MB:  1002.6  :  SatoshiDice Tx: 592.4  (59.1%)

[unclemantis]

I got lucky on a game. I ment to enter 0.10 but entered in 1 instead and i was like NO!!!!! and then I get back like over 2 and change LOL

[dooglus]

See how the blue line goes vertical at the end?  It actually goes up 400 BTC then back down 400 BTC.  I looked at the big bets that caused this and it appears to be a single player:

2012-07-24 05:50:58 lessthan 32000  Bet Amount:  21.50  Outcome: LOSE  Payment:   0.10700000   Profit:  -21.393       Cumulative Profit:  -21.393
2012-07-24 05:51:51 lessthan 32000  Bet Amount:  43.00  Outcome: LOSE  Payment:   0.21450000   Profit:  -42.7855      Cumulative Profit:  -64.1785
2012-07-24 05:52:27 lessthan 32000  Bet Amount:  86.00  Outcome: LOSE  Payment:   0.42950000   Profit:  -85.5705      Cumulative Profit: -149.749
2012-07-24 05:56:39 lessthan 16000  Bet Amount:  86.00  Outcome: LOSE  Payment:   0.42950000   Profit:  -85.5705      Cumulative Profit: -235.3195
2012-07-24 05:54:07 lessthan 32000  Bet Amount: 172.00  Outcome: LOSE  Payment:   0.85950000   Profit: -171.1405      Cumulative Profit: -406.46
2012-07-24 05:58:02 lessthan 16000  Bet Amount: 107.50  Outcome: WIN   Payment: 430.28932000   Profit:  322.78932     Cumulative Profit:  -83.67068
2012-07-24 06:04:53 lessthan 48000  Bet Amount: 250.00  Outcome: WIN   Payment: 334.39083333   Profit:   84.39083333  Cumulative Profit:    0.72015333

He bets thousands of dollars and ends up less than $10 up.  Notice how after losing the 86 BTC bet he bets the same amount at double the odds rather than doubling the bet.  Martingale gets scary quick!

[dooglus]

SatoshiDice profits are almost 800 BTC down since my last update:

Results: 2012-Jul-25 10:19am (up to block 190713)

   Address  Target   Should Win |    #Bets |       Win        |  Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
--------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    10981 |      0 (0.00000) |  10700 |     281 |      61.10 |       0.01 |    17.94 |     61.08 |   0.032
 1dice1Qf4       2      0.00003 |     1037 |      0 (0.00000) |    967 |      70 |      11.14 |       0.00 |     5.58 |     11.14 |   0.016
 1dice2pxm       4      0.00006 |     1512 |      0 (0.00000) |   1479 |      33 |      14.48 |       0.00 |     2.22 |     14.47 |   0.059
 1dice2vQo       8      0.00012 |     1303 |      0 (0.00000) |   1263 |      40 |      18.66 |       0.00 |     4.15 |     18.65 |   0.042
 1dice2WmR      16      0.00024 |     1523 |      0 (0.00000) |   1493 |      30 |      26.05 |       0.02 |     6.60 |     26.02 |   0.107
 1dice2xkj      32      0.00049 |     3652 |      1 (0.00027) |   3640 |      11 |     130.12 |     100.42 |     1.29 |     29.69 |  77.180
 1dice2zdo      64      0.00098 |     5364 |      7 (0.00131) |   5340 |      17 |     226.17 |     121.70 |    55.64 |    104.46 |  53.811
 1dice37Ee     128      0.00195 |     6412 |     15 (0.00236) |   6349 |      48 |    1244.24 |    1148.24 |    40.25 |     95.99 |  92.285
 1dice3jkp     256      0.00391 |     5147 |     24 (0.00468) |   5109 |      14 |     553.11 |     362.14 |    13.11 |    190.97 |  65.473
 1dice4J1m     512      0.00781 |     8172 |     53 (0.00649) |   8114 |       5 |    1604.61 |     823.09 |     9.35 |    781.51 |  51.296
 1dice5wwE    1000      0.01526 |    14588 |    214 (0.01467) |  14371 |       3 |    2417.52 |    2125.60 |     1.80 |    291.92 |  87.925
 1dice61SN    1500      0.02289 |     8296 |    194 (0.02340) |   8096 |       6 |    3122.05 |    3552.96 |    15.00 |   -430.90 | 113.802
 1dice6DPt    2000      0.03052 |    10107 |    316 (0.03127) |   9788 |       3 |    3478.98 |    3206.15 |     9.24 |    272.83 |  92.158
 1dice6gJg    3000      0.04578 |     8195 |    399 (0.04873) |   7789 |       7 |    5030.24 |    6550.35 |    24.99 |  -1520.10 | 130.219
 1dice6GV5    4000      0.06104 |     9079 |    569 (0.06269) |   8507 |       3 |    3180.27 |    2896.93 |    31.20 |    283.33 |  91.091
 1dice6wBx    6000      0.09155 |    16440 |   1546 (0.09408) |  14886 |       8 |    8925.97 |    9067.97 |     7.01 |   -142.00 | 101.591
 1dice6YgE    8000      0.12207 |    33122 |   4110 (0.12411) |  29005 |       7 |    6585.90 |    5832.77 |     0.00 |    753.13 |  88.564
 1dice7EYz   12000      0.18311 |    16604 |   3152 (0.18990) |  13446 |       6 |    6847.82 |    6997.24 |    14.50 |   -149.42 | 102.182
 1dice7fUk   16000      0.24414 |    44500 |  10794 (0.24261) |  33698 |       8 |   15321.65 |   15993.57 |   347.79 |   -671.92 | 104.385
 1dice7W2A   24000      0.36621 |    33157 |  12260 (0.37011) |  20865 |      32 |   13776.12 |   13700.65 |   212.63 |     75.46 |  99.452
 1dice8EMZ   32000      0.48828 |   317884 | 154881 (0.48743) | 162872 |     131 |   99455.79 |   99929.98 |  2173.21 |   -474.19 | 100.477
 1dice97EC   32768      0.50000 |   131626 |  65636 (0.49895) |  65911 |      79 |   49433.29 |   47790.93 |   789.20 |   1642.36 |  96.678
 1dice9wcM   48000      0.73242 |    94745 |  69696 (0.73601) |  24998 |      51 |   78368.91 |   77135.83 |   467.98 |   1233.07 |  98.427
 1dicec9k7   52000      0.79346 |      972 |    778 (0.80041) |    194 |       0 |    1393.07 |    1388.75 |     0.00 |      4.32 |  99.690
 1dicegEAr   56000      0.85449 |      752 |    620 (0.82447) |    132 |       0 |     425.96 |     362.98 |     0.00 |     62.97 |  85.216
 1diceDCd2   60000      0.91553 |      197 |    176 (0.90722) |     18 |       3 |      51.65 |      52.15 |     0.00 |     -0.49 | 100.966
 1dice9wVt   64000      0.97656 |     5969 |   5715 (0.97860) |    125 |     129 |    5036.67 |    4843.84 |   239.20 |    192.82 |  96.172
--------------------------------------------------------------------------------------------------------------------------------------------
                                |   791336 | 331156           | 459155 |    1025 |  306741.65 |  303984.41 |  4489.97 |   2757.23 |  99.101
--------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:       2757.23594710 BTC (0.899%)
Cumulative Fees Paid:         398.36787500 BTC
SD Profit after fees:        2358.86807210 BTC (0.769%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx:  2409837  :  SatoshiDice Tx: 1458898  (60.5%)
Blockchain MB:  1017.2  :  SatoshiDice Tx: 599.6  (58.9%)

[dooglus]

Here's another lucky martingale escape which accounts for the last spike up and back down on the blue line:

2012-07-25 14:51:37 lessthan 16000  Bet Amount:   0.25  Outcome: LOSE  Payment:    0.000750   Profit:   -0.24925   Cumulative Profit: -0.24925 
2012-07-25 14:52:10 lessthan 16000  Bet Amount:   0.50  Outcome: LOSE  Payment:    0.002000   Profit:   -0.498     Cumulative Profit: -0.74725 
2012-07-25 14:52:37 lessthan 16000  Bet Amount:   1.00  Outcome: LOSE  Payment:    0.004500   Profit:   -0.9955    Cumulative Profit: -1.74275 
2012-07-25 14:53:07 lessthan 16000  Bet Amount:   2.00  Outcome: LOSE  Payment:    0.009500   Profit:   -1.9905    Cumulative Profit: -3.73325 
2012-07-25 14:53:38 lessthan 16000  Bet Amount:   4.00  Outcome: LOSE  Payment:    0.019500   Profit:   -3.9805    Cumulative Profit: -7.71375 
2012-07-25 14:54:06 lessthan 16000  Bet Amount:   8.00  Outcome: LOSE  Payment:    0.039500   Profit:   -7.9605    Cumulative Profit: -15.67425
2012-07-25 14:54:32 lessthan 16000  Bet Amount:  16.00  Outcome: LOSE  Payment:    0.079500   Profit:  -15.9205    Cumulative Profit: -31.59475
2012-07-25 14:55:07 lessthan 16000  Bet Amount:  32.00  Outcome: LOSE  Payment:    0.159500   Profit:  -31.8405    Cumulative Profit: -63.43525
2012-07-25 14:55:41 lessthan 16000  Bet Amount:  64.00  Outcome: LOSE  Payment:    0.319500   Profit:  -63.6805    Cumulative Profit: -127.11575
2012-07-25 14:56:17 lessthan 16000  Bet Amount: 128.00  Outcome: LOSE  Payment:    0.639500   Profit: -127.3605    Cumulative Profit: -254.47625
2012-07-25 14:58:10 lessthan 16000  Bet Amount: 219.00  Outcome: LOSE  Payment:    1.094500   Profit: -217.9055    Cumulative Profit: -472.38175
2012-07-25 14:59:36 lessthan 16000  Bet Amount: 219.00  Outcome: WIN   Payment:  876.589924   Profit:  657.589924  Cumulative Profit: 185.208174

[elux]

Here's another lucky martingale escape:

lessthan 16000  Bet Amount:   0.25  Outcome: LOSE  Payment:    0.000750   Profit:   -0.24925   Cumulative Profit: -0.24925 
lessthan 16000  Bet Amount:   0.50  Outcome: LOSE  Payment:    0.002000   Profit:   -0.498     Cumulative Profit: -0.74725 
lessthan 16000  Bet Amount:   1.00  Outcome: LOSE  Payment:    0.004500   Profit:   -0.9955    Cumulative Profit: -1.74275 
lessthan 16000  Bet Amount:   2.00  Outcome: LOSE  Payment:    0.009500   Profit:   -1.9905    Cumulative Profit: -3.73325 
lessthan 16000  Bet Amount:   4.00  Outcome: LOSE  Payment:    0.019500   Profit:   -3.9805    Cumulative Profit: -7.71375 
lessthan 16000  Bet Amount:   8.00  Outcome: LOSE  Payment:    0.039500   Profit:   -7.9605    Cumulative Profit: -15.67425
lessthan 16000  Bet Amount:  16.00  Outcome: LOSE  Payment:    0.079500   Profit:  -15.9205    Cumulative Profit: -31.59475
lessthan 16000  Bet Amount:  32.00  Outcome: LOSE  Payment:    0.159500   Profit:  -31.8405    Cumulative Profit: -63.43525
lessthan 16000  Bet Amount:  64.00  Outcome: LOSE  Payment:    0.319500   Profit:  -63.6805    Cumulative Profit: -127.11575
lessthan 16000  Bet Amount: 128.00  Outcome: LOSE  Payment:    0.639500   Profit: -127.3605    Cumulative Profit: -254.47625
lessthan 16000  Bet Amount: 219.00  Outcome: LOSE  Payment:    1.094500   Profit: -217.9055    Cumulative Profit: -472.38175
lessthan 16000  Bet Amount: 219.00  Outcome: WIN   Payment:  876.589924   Profit:  657.589924  Cumulative Profit: 185.208174

More balls than brains, that one. Kudos upon him.

[gmaxwell]

Here's another lucky martingale escape which accounts for the last spike up and back down on the blue line:

Make sure to chronically the flipside too. The failures, though sad, are also entertaining... and can serve as a cautionary tale for others.

[dooglus]

Sure.  I'm just saying what I see...  Recently the players have been lucky.  Two of the top three "Big Winners" listed on http://www.satoshidice.com/ which lists the biggest wins of all time have been in the last 24 hours.  The biggest loss of all time was also in the last 24 hours, but I documented that 219 BTC loss in the post you replied to - it just happens that the biggest loss ever was part of a winning martingale chain.

In case it's not clear, there is no way of winning long term on satoshi dice without cheating.  There's no way you can turn a positive house edge into a negative house edge simply by adjusting your bet sizes.  I probably assume that everyone knows this when they don't.  That's not to say you can't play and walk away with a profit.  Just that if you keep playing long enough, eventually the house edge will guarantee that you lose.

[dooglus]

Here's another lucky martingale escape:

lessthan 16000  Bet Amount:   0.25  Outcome: LOSE  Payment:    0.000750   Profit:   -0.24925   Cumulative Profit: -0.24925  
lessthan 16000  Bet Amount:   0.50  Outcome: LOSE  Payment:    0.002000   Profit:   -0.498     Cumulative Profit: -0.74725  
lessthan 16000  Bet Amount:   1.00  Outcome: LOSE  Payment:    0.004500   Profit:   -0.9955    Cumulative Profit: -1.74275  
lessthan 16000  Bet Amount:   2.00  Outcome: LOSE  Payment:    0.009500   Profit:   -1.9905    Cumulative Profit: -3.73325  
lessthan 16000  Bet Amount:   4.00  Outcome: LOSE  Payment:    0.019500   Profit:   -3.9805    Cumulative Profit: -7.71375  
lessthan 16000  Bet Amount:   8.00  Outcome: LOSE  Payment:    0.039500   Profit:   -7.9605    Cumulative Profit: -15.67425
lessthan 16000  Bet Amount:  16.00  Outcome: LOSE  Payment:    0.079500   Profit:  -15.9205    Cumulative Profit: -31.59475
lessthan 16000  Bet Amount:  32.00  Outcome: LOSE  Payment:    0.159500   Profit:  -31.8405    Cumulative Profit: -63.43525
lessthan 16000  Bet Amount:  64.00  Outcome: LOSE  Payment:    0.319500   Profit:  -63.6805    Cumulative Profit: -127.11575
lessthan 16000  Bet Amount: 128.00  Outcome: LOSE  Payment:    0.639500   Profit: -127.3605    Cumulative Profit: -254.47625
lessthan 16000  Bet Amount: 219.00  Outcome: LOSE  Payment:    1.094500   Profit: -217.9055    Cumulative Profit: -472.38175
lessthan 16000  Bet Amount: 219.00  Outcome: WIN   Payment:  876.589924   Profit:  657.589924  Cumulative Profit: 185.208174

More balls than brains, that one. Kudos upon him.

It looks like the same guy was responsible for the other big loss of the last day:

2012-07-25 03:11:26 lessthan 16000  Bet Amount:   4.00  Outcome: LOSE       Payment:   0.019500   Profit:   -3.9805    Cumulative Profit:    -3.9805  
2012-07-25 03:11:54 lessthan 16000  Bet Amount:   8.00  Outcome: LOSE       Payment:   0.039500   Profit:   -7.9605    Cumulative Profit:   -11.941  
2012-07-25 03:13:31 lessthan 16000  Bet Amount:  16.00  Outcome: LOSE       Payment:   0.079500   Profit:  -15.9205    Cumulative Profit:   -27.8615  
2012-07-25 03:14:10 lessthan 16000  Bet Amount:  32.00  Outcome: LOSE       Payment:   0.159500   Profit:  -31.8405    Cumulative Profit:   -59.702  
2012-07-25 03:14:47 lessthan 16000  Bet Amount:  64.00  Outcome: LOSE       Payment:   0.319500   Profit:  -63.6805    Cumulative Profit:  -123.3825  
2012-07-25 03:15:30 lessthan 16000  Bet Amount: 128.00  Outcome: LOSE       Payment:   0.639500   Profit: -127.3605    Cumulative Profit:  -250.743  
2012-07-25 03:16:11 lessthan 16000  Bet Amount: 250.00  Outcome: ABOVE MAX  Payment: 250.000000   Profit:    0.0       Cumulative Profit:  -250.743  
2012-07-25 03:17:14 lessthan 16000  Bet Amount: 219.00  Outcome: WIN        Payment: 876.589924   Profit:  657.589924  Cumulative Profit:   406.846924

Very similar betting pattern, but he didn't know about the 219 BTC max bet at this point.

[traderjoe]

...
In case it's not clear, there is no way of winning long term on satoshi dice without cheating.  There's no way you can turn a positive house edge into a negative house edge simply by adjusting your bet sizes.  I probably assume that everyone knows this when they don't.  That's not to say you can't play and walk away with a profit.  Just that if you keep playing long enough, eventually the house edge will guarantee that you lose.

True, but don't you think that if too many of the players bet this way, the house is more likely to bust?  The table bet limits are the house's friend.