Satoshi Dice -- Statistical Analysis

[dooglus]

He's the only one that can promise it as he is the only one speaking for the majority shareholder, who is the only one with the power to print more shares.

Shareholder dilution is the best way to take away value from shareholders. Similar to printing money.

This isn't about dilution.  There are 100 million shares, of which 13 million are currently available.  When this went up from 10 million to 13 million there was no dilution.  Each share still gets one 100 millionth of the monthly profits as a dividend.

Could it be that actual profit is considerably higher than expected profit because gamblers on average stop more when they have been exceptionally unlucky?

I don't know, and it hurts my head to think about.

Everything I know about probability tells me that there's no way of changing the expected profit by changing your betting strategy.  "Quitting while you're behind" is a betting strategy, and so shouldn't affect the house edge.

If it did then we could beat the house by all "quitting while we're ahead", right?

As far as the bottom line is concerned, there's no difference between one big player and another.  It doesn't matter if each individual player stops when they are down a lot; it's just a stream of bets, each of which is paid out 1.9% less than the 'fair' amount.  That has to lead to a 1.9% profit overall I think in the long run.

[1714205130]

1714205130

[1714205130]

1714205130

[1714205130]

1714205130

[1714205130]

1714205130

[1714205130]

1714205130

[organofcorti]

He's the only one that can promise it as he is the only one speaking for the majority shareholder, who is the only one with the power to print more shares.

Shareholder dilution is the best way to take away value from shareholders. Similar to printing money.

This isn't about dilution.  There are 100 million shares, of which 13 million are currently available.  When this went up from 10 million to 13 million there was no dilution.  Each share still gets one 100 millionth of the monthly profits as a dividend.

Could it be that actual profit is considerably higher than expected profit because gamblers on average stop more when they have been exceptionally unlucky?

I don't know, and it hurts my head to think about.

Me too.

Everything I know about probability tells me that there's no way of changing the expected profit by changing your betting strategy.  "Quitting while you're behind" is a betting strategy, and so shouldn't affect the house edge.

If it did then we could beat the house by all "quitting while we're ahead", right?

Since this is all a bit fuzzy and I haven't the foggiest idea how to even approach analysing it, I'm probably wrong, but here goes:

The difference is that  "quitting while we're ahead" means the first time you're ahead, which will probably happen early on with only a small amount bet gained.

"Quitting while you're behind" in this case means you've lost all you can afford to lose, which won't necessarily be a small amount.

[dooglus]

The difference is that  "quitting while we're ahead" means the first time you're ahead, which will probably happen early on with only a small amount bet gained.

"Quitting while you're behind" in this case means you've lost all you can afford to lose, which won't necessarily be a small amount.

The other difference is that the "ahead" is a state that may never happen for some players, if they lose their first few bets and never recover those losses.  On the other hand, given the 1.9% house edge, the "behind" state will happen for all players if they just keep playing long enough.  In fact so will the "lost it all" state if they play enough times.


For the purposes of analysis, imagine there's an endless line of players who take on SatoshiDice one at a time, like in a kung-fu movie.  Each one starts with 100 BTC and plays until he has lost it all, then quits forever.

We can say that after the Nth player has finished playing, the house profits will be 100*N BTC.

And we can also say that on average each player bets a total of 100 / (1.9/100) = 5263.16 BTC.

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.

[Bugpowder]

The difference is that  "quitting while we're ahead" means the first time you're ahead, which will probably happen early on with only a small amount bet gained.

"Quitting while you're behind" in this case means you've lost all you can afford to lose, which won't necessarily be a small amount.

The other difference is that the "ahead" is a state that may never happen for some players, if they lose their first few bets and never recover those losses.  On the other hand, given the 1.9% house edge, the "behind" state will happen for all players if they just keep playing long enough.  In fact so will the "lost it all" state if they play enough times.


For the purposes of analysis, imagine there's an endless line of players who take on SatoshiDice one at a time, like in a kung-fu movie.  Each one starts with 100 BTC and plays until he has lost it all, then quits forever.

We can say that after the Nth player has finished playing, the house profits will be 100*N BTC.

And we can also say that on average each player bets a total of 100 / (1.9/100) = 5263.16 BTC.

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.

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?

[dooglus]

Total of 72 bets unaccounted for.

Results: 2013-May-17 01:26am (up to block 236579)

   Address  Target   Should Win |    #Bets |        Win        |   Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
----------------------------------------------------------------------------------------------------------------------------------------------
 1dice9wVt   64000      0.97656 |    23892 |   21575 (0.97859) |     472 |    1845 |   25519.74 |   25093.60 |   241.32 |    426.13 |  98.330
 1diceDCd2   60000      0.91553 |   146224 |  133037 (0.91584) |   12226 |     961 |   79528.53 |   78326.53 |     2.35 |   1202.00 |  98.489
 1dicegEAr   56000      0.85449 |    93421 |   79393 (0.85714) |   13232 |     796 |   85265.91 |   84503.41 |   401.34 |    762.49 |  99.106
 1dicec9k7   52000      0.79346 |    98423 |   77561 (0.79385) |   20141 |     721 |   67289.19 |   65623.23 |  1188.22 |   1665.96 |  97.524
 1dice9wcM   48000      0.73242 |   429305 |  314740 (0.73483) |  113574 |     991 |  315344.18 |  307876.47 |  6173.26 |   7467.71 |  97.632
 1dice97EC   32768      0.50000 |   771150 |  384434 (0.49970) |  384888 |    1828 |  718138.06 |  700382.60 |  6523.96 |  17755.45 |  97.528
 1dice8EMZ   32000      0.48828 |  1291140 |  629198 (0.48809) |  659902 |    2040 |  932138.91 |  913684.89 |  2926.52 |  18454.02 |  98.020
 1dice7W2A   24000      0.36621 |   305339 |  111900 (0.36734) |  192722 |     717 |  598625.00 |  589916.21 |  1013.80 |   8708.78 |  98.545
 1dice7fUk   16000      0.24414 |   312367 |   76036 (0.24403) |  235553 |     778 |  382593.45 |  368618.95 |  2323.25 |  13974.50 |  96.347
 1dice7EYz   12000      0.18311 |   130321 |   23800 (0.18357) |  105849 |     672 |  181455.31 |  183283.92 |  3315.40 |  -1828.61 | 101.008
 1dice6YgE    8000      0.12207 |   230828 |   28293 (0.12292) |  201890 |     645 |  102924.34 |  102290.22 |   101.12 |    634.11 |  99.384
 1dice6wBx    6000      0.09155 |    61364 |    5559 (0.09142) |   55248 |     557 |   17545.65 |   19190.80 |     7.68 |  -1645.15 | 109.376
 1dice6GV5    4000      0.06104 |    40351 |    2480 (0.06211) |   37447 |     424 |    8040.38 |    7961.74 |    31.47 |     78.64 |  99.022
 1dice6gJg    3000      0.04578 |    32894 |    1503 (0.04641) |   30881 |     510 |   10211.00 |   11231.07 |    25.47 |  -1020.07 | 109.990
 1dice6DPt    2000      0.03052 |   102276 |    3110 (0.03054) |   98712 |     454 |   44173.55 |   39737.82 |     9.58 |   4435.72 |  89.958
 1dice61SN    1500      0.02289 |    27903 |     641 (0.02331) |   26861 |     401 |    8123.51 |    8715.41 |    15.23 |   -591.89 | 107.286
 1dice5wwE    1000      0.01526 |   159591 |    2453 (0.01544) |  156458 |     680 |   43689.05 |   37669.12 |     2.65 |   6019.93 |  86.221
 1dice4J1m     512      0.00781 |    41299 |     323 (0.00797) |   40179 |     797 |    8518.85 |    6964.25 |    10.36 |   1554.60 |  81.751
 1dice3jkp     256      0.00391 |    27822 |     117 (0.00427) |   27280 |     425 |    6113.17 |    8664.70 |    13.31 |  -2551.53 | 141.738
 1dice37Ee     128      0.00195 |    22816 |      42 (0.00188) |   22255 |     519 |    2449.17 |    1583.25 |    48.58 |    865.92 |  64.644
 1dice2zdo      64      0.00098 |    31451 |      35 (0.00113) |   30861 |     555 |    2210.00 |    1740.51 |   117.13 |    469.48 |  78.756
 1dice2xkj      32      0.00049 |    30954 |      13 (0.00043) |   30484 |     457 |    1680.88 |    1831.02 |     1.60 |   -150.14 | 108.932
 1dice2WmR      16      0.00024 |    25000 |       5 (0.00020) |   24504 |     491 |     889.21 |     606.68 |    22.21 |    282.53 |  68.226
 1dice2vQo       8      0.00012 |    25803 |       5 (0.00020) |   25244 |     554 |     617.39 |     433.94 |    14.66 |    183.45 |  70.286
 1dice2pxm       4      0.00006 |    15021 |       2 (0.00014) |   14485 |     534 |     215.33 |     320.40 |    18.46 |   -105.07 | 148.796
 1dice1Qf4       2      0.00003 |    13248 |       0 (0.00000) |   12663 |     585 |     181.83 |       0.49 |    28.56 |    181.33 |   0.275
 1dice1e6p       1      0.00002 |   145368 |       3 (0.00002) |  142844 |    2521 |    1846.60 |    3843.72 |   174.69 |  -1997.12 | 208.151
----------------------------------------------------------------------------------------------------------------------------------------------
           small (bets < 4 BTC) |  4510561 | 1839886           | 2648511 |   22164 |  878419.17 |  862991.28 |   346.09 |  15427.89 |  98.244
            big (bets >= 4 BTC) |   125010 |   56372           |   68344 |     294 | 2766909.13 | 2707103.80 | 24406.21 |  59805.33 |  97.839
----------------------------------------------------------------------------------------------------------------------------------------------
                                |  4635571 | 1896258           | 2716855 |   22458 | 3645328.31 | 3570095.08 | 24752.31 |  75233.22 |  97.936
----------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:      75233.22431411 BTC (2.064%)
Cumulative Fees Paid:        3878.96977500 BTC
SD Profit after fees:       71354.25453911 BTC (1.957%)
Pending Liabilities:          -15.76743849 BTC
Final SD Profit:            71370.02197760 BTC (1.958%)
Profit This Month:           -233.38636702 BTC
----
Since Satoshi Dice started, there have been:
Blockchain Tx: 15087441  :  SatoshiDice Tx:  8512696  (56.4%)
Blockchain MB:   6624.7  :  SatoshiDice MB:   3511.4  (53.0%)

[MPOE-PR]

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?

Same as in the neighboring country, I would guess? (Many boys just after a war, many girls just before one).

[Zaih]

The difference is that  "quitting while we're ahead" means the first time you're ahead, which will probably happen early on with only a small amount bet gained.

"Quitting while you're behind" in this case means you've lost all you can afford to lose, which won't necessarily be a small amount.

The other difference is that the "ahead" is a state that may never happen for some players, if they lose their first few bets and never recover those losses.  On the other hand, given the 1.9% house edge, the "behind" state will happen for all players if they just keep playing long enough.  In fact so will the "lost it all" state if they play enough times.


For the purposes of analysis, imagine there's an endless line of players who take on SatoshiDice one at a time, like in a kung-fu movie.  Each one starts with 100 BTC and plays until he has lost it all, then quits forever.

We can say that after the Nth player has finished playing, the house profits will be 100*N BTC.

And we can also say that on average each player bets a total of 100 / (1.9/100) = 5263.16 BTC.

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.

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?

Exactly the same still. Probability is a continuum regardless of how you go about it.

[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.