Satoshi Dice -- Statistical Analysis

[nimda]

Schools need to teach more probability and formal logic.

Relevant analogy :
If there is a country where every husband and wife decides to have children until they have a boy, what is the distribution of boys vs. girls in the country?

This is a great analogy. Let's see if a written approach can help:

Separating the trial-end condition from the trials themselves, we know that on average, the distribution should be equal. However, if it's not inherently obvious that the trial-end condition doesn't affect the probability, consider this:

Half of the pairs will have a boy on the first try.
The other half will have a girl on the first try.

Now, set aside the first try and the male births. Start over, ignoring the first birth, with all of the couples who had girls.
Half of them will have a boy on this try.
The other half will have a girl on this try.

Wash, rinse, repeat.

50% of the country has one male child.
25% of the country has one male and one female child.
12.5% ... 2 females, one male
6.25% ... 3 females, one male
3.125% ... 4 females, one male
etc etc

If there were 100 couples, it should be obvious from the definition that there would be 100 boys, since each couple has a boy. But how many girls would there be? On average, 100.

Mathematically:
The population of females is the sum of the sequence (n-1)/(2^n) from n=1 to infinity
The population of males is the sum of the sequence (1)/(2^n) from n=1 to infinity
I can go into more detail, but both of these sequences converge, and both of them are equal to 1. (Thus each parent will have, on average, 2 children.)

What most likely throws people off is the different "speeds" at which these functions change. Plug them into Wolfram Alpha and compare the graphs.

[mootinator]

I think I've satisfied myself (though probably nobody else) that people's tendencies to quit only when they've lost their bankroll doesn't affect the house's percentage profits, and that the current better-than-expected profits are merely the result of some good house luck earlier this year.  Even if that whale was always intending to keep playing until he lost it all, that process could have taken him 10 times longer than it did.  The house profits would be the same as they are now, but they would then be less than expected instead of being more than expected as they currently are.

Put another way, even if a player makes N bets or until the player is ruined, probability really doesn't care that the next bet comes from someone else when someone leaves early.

[dooglus]

50% of the country has one male child.
25% of the country has one male and one female child.
12.5% ... 2 females, one male
6.25% ... 3 females, one male
3.125% ... 4 females, one male
etc etc

Or you can look at it this way:

50% of the country has at least 1 girl - contributing P/2 to the girl population
25% of the country has a 2nd girl - contributing P/4 to the girl population
12.5% of the country has a 3rd girl - contributing P/8 to the girl population
...

so the total number of girls is P(1/2+1/4+1/8+ ...) = P.

Anyway, what do you want first, the good news or the bad news? 

Fri May 17 15:01:44 2013 <12000 236623 c37a6816b2fa5dc7:0  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    25.00000000 WIN    108.38153333
Fri May 17 15:26:47 2013 <12000 236626 3f464ab7040e78d9:1  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    25.00000000 WIN    108.38153333
Fri May 17 16:35:45 2013 <8000  236631 b7761f307c676def:0  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    17.00000000 WIN    119.00666400
Fri May 17 16:35:45 2013 <1000  236631 0f97c0d6c98c797f:1  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET     5.00000000 WIN    314.84068000
Fri May 17 16:39:35 2013 <2000  236632 4e2a13023722544b:2  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    14.00000000 WIN    433.81195200
Fri May 17 17:06:35 2013 <12000 236637 798658472ee15616:0  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    26.00000000 WIN    112.71679466
Fri May 17 17:30:27 2013 <8000  236639 b374402235e391ff:0  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    23.00000000 WIN    161.00901600
Fri May 17 17:51:42 2013 <1000  236643 6c67df9b6225abb6:2  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    13.00000000 WIN    818.58576800
Fri May 17 18:05:28 2013 <8000  236645 0a54ac3736a07159:1  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    16.00000000 WIN    112.00627200
Fri May 17 18:12:46 2013 <8000  236646 d1031eb2e84bdecb:0  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    23.00000000 WIN    161.00901600
Fri May 17 19:00:12 2013 <8000  236652 4188dd37d162de6c:1  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET    24.00000000 WIN    168.00940800

The whale seems to be back...  and he seems to be winning...

Total of 85 bets unaccounted for.

Results: 2013-May-17 01:29pm (up to block 236659)

   Address  Target   Should Win |    #Bets |        Win        |   Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
----------------------------------------------------------------------------------------------------------------------------------------------
 1dice9wVt   64000      0.97656 |    23910 |   21591 (0.97856) |     473 |    1846 |   25524.46 |   25098.31 |   241.33 |    426.15 |  98.330
 1diceDCd2   60000      0.91553 |   146711 |  133475 (0.91579) |   12273 |     963 |   79705.05 |   78499.63 |     2.36 |   1205.41 |  98.488
 1dicegEAr   56000      0.85449 |    93700 |   79630 (0.85713) |   13273 |     797 |   85374.11 |   84617.71 |   401.34 |    756.39 |  99.114
 1dicec9k7   52000      0.79346 |    98514 |   77626 (0.79379) |   20166 |     722 |   67301.27 |   65631.73 |  1188.23 |   1669.54 |  97.519
 1dice9wcM   48000      0.73242 |   430008 |  315251 (0.73482) |  113765 |     992 |  315785.33 |  307996.85 |  6173.26 |   7788.47 |  97.534
 1dice97EC   32768      0.50000 |   772758 |  385262 (0.49974) |  385664 |    1832 |  719371.42 |  701984.45 |  6523.96 |  17386.96 |  97.583
 1dice8EMZ   32000      0.48828 |  1293794 |  630521 (0.48811) |  661227 |    2046 |  933174.40 |  914377.89 |  2926.55 |  18796.50 |  97.986
 1dice7W2A   24000      0.36621 |   305830 |  112078 (0.36733) |  193035 |     717 |  599165.15 |  590413.84 |  1013.80 |   8751.31 |  98.539
 1dice7fUk   16000      0.24414 |   313635 |   76351 (0.24405) |  236503 |     781 |  383081.39 |  369005.12 |  2323.28 |  14076.26 |  96.326
 1dice7EYz   12000      0.18311 |   130592 |   23854 (0.18361) |  106065 |     673 |  181975.53 |  183778.51 |  3315.41 |  -1802.97 | 100.991
 1dice6YgE    8000      0.12207 |   231199 |   28334 (0.12290) |  202220 |     645 |  103604.91 |  103293.64 |   101.12 |    311.27 |  99.700
 1dice6wBx    6000      0.09155 |    61572 |    5580 (0.09145) |   55434 |     558 |   17550.89 |   19193.48 |     7.69 |  -1642.58 | 109.359
 1dice6GV5    4000      0.06104 |    40360 |    2481 (0.06212) |   37455 |     424 |    8040.49 |    7961.90 |    31.47 |     78.58 |  99.023
 1dice6gJg    3000      0.04578 |    32927 |    1505 (0.04643) |   30911 |     511 |   10211.32 |   11231.50 |    25.47 |  -1020.18 | 109.991
 1dice6DPt    2000      0.03052 |   102411 |    3111 (0.03051) |   98846 |     454 |   44683.07 |   40188.11 |     9.58 |   4494.96 |  89.940
 1dice61SN    1500      0.02289 |    27925 |     641 (0.02329) |   26883 |     401 |    8123.90 |    8715.41 |    15.23 |   -591.51 | 107.281
 1dice5wwE    1000      0.01526 |   159784 |    2456 (0.01544) |  156648 |     680 |   43914.82 |   38822.22 |     2.65 |   5092.59 |  88.403
 1dice4J1m     512      0.00781 |    41316 |     323 (0.00797) |   40196 |     797 |    8519.06 |    6964.25 |    10.36 |   1554.80 |  81.749
 1dice3jkp     256      0.00391 |    27831 |     117 (0.00427) |   27288 |     426 |    6113.34 |    8664.70 |    13.31 |  -2551.36 | 141.734
 1dice37Ee     128      0.00195 |    22825 |      42 (0.00188) |   22264 |     519 |    2449.26 |    1583.25 |    48.58 |    866.01 |  64.642
 1dice2zdo      64      0.00098 |    31455 |      35 (0.00113) |   30865 |     555 |    2210.04 |    1740.51 |   117.13 |    469.52 |  78.755
 1dice2xkj      32      0.00049 |    30972 |      13 (0.00043) |   30502 |     457 |    1681.11 |    1831.02 |     1.60 |   -149.91 | 108.917
 1dice2WmR      16      0.00024 |    25002 |       5 (0.00020) |   24505 |     492 |     889.22 |     606.68 |    22.21 |    282.54 |  68.226
 1dice2vQo       8      0.00012 |    25807 |       5 (0.00020) |   25248 |     554 |     617.43 |     433.94 |    14.66 |    183.49 |  70.281
 1dice2pxm       4      0.00006 |    15034 |       2 (0.00014) |   14497 |     535 |     215.45 |     320.41 |    18.47 |   -104.95 | 148.714
 1dice1Qf4       2      0.00003 |    13265 |       0 (0.00000) |   12680 |     585 |     182.00 |       0.50 |    28.56 |    181.50 |   0.275
 1dice1e6p       1      0.00002 |   145698 |       3 (0.00002) |  143171 |    2524 |    1850.23 |    3843.74 |   174.71 |  -1993.51 | 207.744
----------------------------------------------------------------------------------------------------------------------------------------------
           small (bets < 4 BTC) |  4519587 | 1843861           | 2653534 |   22192 |  879377.92 |  863939.64 |   346.23 |  15438.28 |  98.244
            big (bets >= 4 BTC) |   125248 |   56431           |   68523 |     294 | 2771936.85 | 2712859.79 | 24406.21 |  59077.06 |  97.869
----------------------------------------------------------------------------------------------------------------------------------------------
                                |  4644835 | 1900292           | 2722057 |   22486 | 3651314.78 | 3576799.43 | 24752.45 |  74515.35 |  97.959
----------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:      74515.35034495 BTC (2.041%)
Cumulative Fees Paid:        3888.77077500 BTC
SD Profit after fees:       70626.57956995 BTC (1.934%)
Pending Liabilities:          321.82134594 BTC
Final SD Profit:            70304.75822401 BTC (1.925%)
Profit This Month:          -1298.65012061 BTC
----
Since Satoshi Dice started, there have been:
Blockchain Tx: 15118526  :  SatoshiDice Tx:  8530326  (56.4%)
Blockchain MB:   6638.5  :  SatoshiDice MB:   3518.7  (53.0%)

[fredtrader]

It's going to be a bad month for dividends

[Korbman]

It's going to be a bad month for dividends

Works for me. People will start panic selling as they always do and give me a chance to buy in super low

[fredtrader]

It's going to be a bad month for dividends

Works for me. People will start panic selling as they always do and give me a chance to buy in super low

Yeah true but the overall amount bet also seems to be going down. I think the site has lost quite a bit of its original popularity.

[nimda]

50% of the country has one male child.
25% of the country has one male and one female child.
12.5% ... 2 females, one male
6.25% ... 3 females, one male
3.125% ... 4 females, one male
etc etc

Or you can look at it this way:

50% of the country has at least 1 girl - contributing P/2 to the girl population
25% of the country has a 2nd girl - contributing P/4 to the girl population
12.5% of the country has a 3rd girl - contributing P/8 to the girl population
...

so the total number of girls is P(1/2+1/4+1/8+ ...) = P.

Or, you can fall back to the axioms: there are n trials, and on average n/2 of them will be girls. The trial conditions are irrelevant to the probability. This is just a bit hard to grok.

[organofcorti]

If anyone's still not getting the boy / girl problem, it's a Bernoulli trial, just like pooled bitcoin mining.

Think of it this way - a family keeps submitting babies until it submits a boy and the family is solved (the difference is that in pooled bitcoin mining all shares that are submitted are counted, not just the girls - so pooled bitcoin mining is a "shifted" geometric distribution).

Once you understand that, then you can estimate the various parameters of the number of girls submitted per solved family.

So, without providing proof,

p = 0.5     (a 50 - 50 chance of boy or girl per submission attempt, or to push the mining analogy way too far, baby mining difficulty = 2)

mean for girls submitted per solved family = (1-p)/p = 1

The reason that I think people might be confused is that since there will be far more families without girls than for any other number "n' of girls per family, the median girls per family is zero. (As dooglus points out below, half the families should have no girls).

median for girls submitted per solved family = ceiling(-1/log2(1-p)) - 1 = 0

and as you'd expect the variance is quite large:

variance for girls submitted per solved family = (1-p)/p^2 = 2

Good luck with your pooled baby mining.

[dooglus]

there will be far more families without girls than with

Isn't it equal?

Half the families have a boy first and quit while they're ahead, and so have no girls.  The other half have a girl first, and so have a girl.

[organofcorti]

there will be far more families without girls than with

Isn't it equal?

Half the families have a boy first and quit while they're ahead, and so have no girls.  The other half have a girl first, and so have a girl.

That's the mean. If half the families have no girls, of the remainder less than half have one girl. Some have two or three. The median is the "midway" number, the number that has an equal number of results on either side. Since half the families have no girls,  the median number is 0.

Edit: Oh, I see what you mean. That was a dumb thing to write, eh? I'll fix it .... and fixed.

[MPOE-PR]

Good luck with your pooled baby mining.

I lolled.

[dooglus]

If half the families have no girls, of the remainder less than half have one girl. Some have two or three. The median is the "midway" number, the number that has an equal number of results on either side. Since half the families have no girls,  the median number is 0.

Edit: Oh, I see what you mean. That was a dumb thing to write, eh? I'll fix it .... and fixed.

I still don't like it.  If half the families have 0 girls, and half the families have 1 or more, then isn't the median 0.5?  The median is half-way between the highest value in the lowest half (0) and the lowest value in the highest half (1).

[organofcorti]

If half the families have no girls, of the remainder less than half have one girl. Some have two or three. The median is the "midway" number, the number that has an equal number of results on either side. Since half the families have no girls,  the median number is 0.

Edit: Oh, I see what you mean. That was a dumb thing to write, eh? I'll fix it .... and fixed.

I still don't like it.  If half the families have 0 girls, and half the families have 1 or more, then isn't the median 0.5?  The median is half-way between the highest value in the lowest half (0) and the lowest value in the highest half (1).

The way I work it out:

The median is when the CDF = 0.5 (ie 50th percentile).

The probability of any family being complete after the next birth:

p * (1 - p)^(n + 1)

where p = prob(girl) and n = nth child


so the CDF P(X > x) will be sum[p * (1 - p)^(n + 1)] from 0 to x

Using wolfram alpha, you get p (1-p)^x-(1-p)^x+1, or 1 - (1 - p)^(x + 1).


From here you need to calculate the quantile function, and I'm going to let wolfram alpha do the heavy lifting again:

P = 1 - (1 - p)^(x + 1), So
x = log((P-1)/(p-1))/log(1-p)

In this case, we want to find the median, where P(X > x) = 0.5

log((P-1)/(p-1))/log(1-p) = log(1)/log(0.5) = 0

So the median value is zero.

However, you could still be correct. If you read wikipedia, there's a formula for the median of a geometric distribution, and a statement that if part of the result is an integer then the median is "not unique". In this case that's true, but I don't know why or what it means.

[tulkos]

Anyway, what do you want first, the good news or the bad news? 

So what was the good news?

[romerun]

whale is back

[RationalSpeculator]

The whale seems to be back...

That is great news.

Historically this meant expected profit tripled in only 1 month. Let's see how it plays out now.

[dooglus]

So what was the good news?

good news: whale is back
bad news: he's kicking our ass

[RationalSpeculator]

So what was the good news?

good news: whale is back
bad news: he's kicking our ass

Is that really bad news?

True it can destroy the dividend temporarily but at the same time the whale is raising expected profit still dramatically.

Expected profit is much more important than actual profit, as expected profit tells the true story about future profits.

The whale being back is very valuable confirmation for me that whales are not a one time event but will continue to come.

Him having temporary luck is exactly why they will continue to come back. It just feels so good and that's exactly what sdice offers. A temporary high!  
 

[dooglus]

Is that really bad news?

Only if keeps happening...    But in general I think casinos like it when players win.  It's good publicity, and they generally win it back eventually anyway.

Total of 49 bets unaccounted for.

Results: 2013-May-19 02:08pm (up to block 236968)

   Address  Target   Should Win |    #Bets |        Win        |   Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
----------------------------------------------------------------------------------------------------------------------------------------------
 1dice9wVt   64000      0.97656 |    23969 |   21643 (0.97857) |     474 |    1852 |   25549.25 |   25123.20 |   241.34 |    426.05 |  98.332
 1diceDCd2   60000      0.91553 |   147630 |  134299 (0.91574) |   12357 |     974 |   80040.59 |   78838.05 |     2.42 |   1202.53 |  98.498
 1dicegEAr   56000      0.85449 |    94484 |   80275 (0.85685) |   13411 |     798 |   85913.81 |   85139.93 |   401.34 |    773.88 |  99.099
 1dicec9k7   52000      0.79346 |    99286 |   78254 (0.79395) |   20309 |     723 |   67619.30 |   65924.91 |  1188.23 |   1694.38 |  97.494
 1dice9wcM   48000      0.73242 |   432513 |  317051 (0.73475) |  114459 |    1003 |  316410.10 |  308626.83 |  6173.32 |   7783.27 |  97.540
 1dice97EC   32768      0.50000 |   780259 |  389029 (0.49977) |  389386 |    1844 |  723346.00 |  705858.66 |  6524.01 |  17487.33 |  97.582
 1dice8EMZ   32000      0.48828 |  1302037 |  634575 (0.48814) |  665409 |    2053 |  934742.92 |  915859.73 |  2926.58 |  18883.19 |  97.980
 1dice7W2A   24000      0.36621 |   307358 |  112674 (0.36745) |  193964 |     720 |  599799.64 |  591078.71 |  1013.81 |   8720.93 |  98.546
 1dice7fUk   16000      0.24414 |   316094 |   76952 (0.24405) |  238358 |     784 |  384195.33 |  369780.11 |  2323.31 |  14415.22 |  96.248
 1dice7EYz   12000      0.18311 |   131664 |   24067 (0.18373) |  106921 |     676 |  183057.56 |  184589.80 |  3315.43 |  -1532.23 | 100.837
 1dice6YgE    8000      0.12207 |   232567 |   28512 (0.12294) |  203406 |     649 |  104620.51 |  104217.37 |   101.13 |    403.14 |  99.615
 1dice6wBx    6000      0.09155 |    62279 |    5643 (0.09143) |   56078 |     558 |   17617.03 |   19358.76 |     7.69 |  -1741.73 | 109.887
 1dice6GV5    4000      0.06104 |    40426 |    2485 (0.06212) |   37517 |     424 |    8048.51 |    7962.82 |    31.47 |     85.68 |  98.935
 1dice6gJg    3000      0.04578 |    32973 |    1505 (0.04636) |   30957 |     511 |   10215.53 |   11231.52 |    25.47 |  -1015.99 | 109.946
 1dice6DPt    2000      0.03052 |   103040 |    3130 (0.03051) |   99453 |     457 |   45474.75 |   40586.88 |     9.59 |   4887.86 |  89.251
 1dice61SN    1500      0.02289 |    27968 |     642 (0.02329) |   26925 |     401 |    8126.68 |    8715.85 |    15.23 |   -589.17 | 107.250
 1dice5wwE    1000      0.01526 |   160513 |    2465 (0.01542) |  157364 |     684 |   44387.53 |   39597.30 |     2.67 |   4790.22 |  89.208
 1dice4J1m     512      0.00781 |    41456 |     324 (0.00797) |   40335 |     797 |    8525.02 |    6965.53 |    10.36 |   1559.49 |  81.707
 1dice3jkp     256      0.00391 |    27858 |     117 (0.00427) |   27315 |     426 |    6118.23 |    8664.72 |    13.31 |  -2546.49 | 141.621
 1dice37Ee     128      0.00195 |    22864 |      42 (0.00188) |   22302 |     520 |    2450.38 |    1583.25 |    48.58 |    867.13 |  64.612
 1dice2zdo      64      0.00098 |    31524 |      35 (0.00113) |   30931 |     558 |    2211.46 |    1740.52 |   117.14 |    470.93 |  78.705
 1dice2xkj      32      0.00049 |    31002 |      13 (0.00043) |   30532 |     457 |    1682.16 |    1831.03 |     1.60 |   -148.86 | 108.850
 1dice2WmR      16      0.00024 |    25027 |       5 (0.00020) |   24529 |     493 |     889.90 |     606.68 |    22.21 |    283.22 |  68.174
 1dice2vQo       8      0.00012 |    25844 |       5 (0.00020) |   25285 |     554 |     618.29 |     433.94 |    14.66 |    184.34 |  70.185
 1dice2pxm       4      0.00006 |    15072 |       2 (0.00014) |   14534 |     536 |     216.63 |     320.41 |    18.48 |   -103.77 | 147.904
 1dice1Qf4       2      0.00003 |    13293 |       0 (0.00000) |   12708 |     585 |     182.73 |       0.50 |    28.56 |    182.23 |   0.276
 1dice1e6p       1      0.00002 |   146695 |       3 (0.00002) |  144143 |    2549 |    1866.02 |    3843.83 |   175.81 |  -1977.80 | 205.991
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           small (bets < 4 BTC) |  4549797 | 1857124           | 2670381 |   22292 |  884297.63 |  868709.20 |   347.68 |  15588.43 |  98.237
            big (bets >= 4 BTC) |   125898 |   56623           |   68981 |     294 | 2779628.36 | 2719771.76 | 24406.21 |  59856.59 |  97.847
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                                |  4675695 | 1913747           | 2739362 |   22586 | 3663925.99 | 3588480.97 | 24753.90 |  75445.02 |  97.941
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SD Profit before fees:      75445.02892822 BTC (2.059%)
Cumulative Fees Paid:        3921.12217500 BTC
SD Profit after fees:       71523.90675322 BTC (1.952%)
Pending Liabilities:           -0.21907916 BTC
Final SD Profit:            71524.12583238 BTC (1.952%)
Profit This Month:            -79.28251224 BTC
----
Since Satoshi Dice started, there have been:
Blockchain Tx: 15219484  :  SatoshiDice Tx:  8587114  (56.4%)
Blockchain MB:   6683.8  :  SatoshiDice MB:   3542.1  (53.0%)

[dooglus]

Recent wins making over 100 BTC profit:

Sat May 18 00:36:01 2013 <2000  236686 2c9394f2cdc85fc7:0  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET     6.00000000 WIN    185.91940800
Sat May 18 03:17:03 2013 <8000  236696 f37bdd3e26be6a6b:2  1Cp7czVeWweLwf14evirmAvtvGJvD5AEyu BET    17.00000000 WIN    119.00666400
Sat May 18 04:17:00 2013 <1000  236703 22ee1ad305de5bcf:4  1Cp7czVeWweLwf14evirmAvtvGJvD5AEyu BET     5.00000000 WIN    314.84068000
Sat May 18 05:35:13 2013 <8000  236708 f9791ea0aeaf7e28:2  1Cp7czVeWweLwf14evirmAvtvGJvD5AEyu BET    17.00000000 WIN    119.00666400
Sat May 18 13:02:44 2013 <2000  236752 22f4a5d76673e4bd:3  1BDDDQ8VhXPF11m2rDoVJXj1BgWPazEttz BET     6.00000000 WIN    185.91940800