♠ Monday, May 17, 2004
A Note on Variance
A thought just occurred to me, although it's surprising that it took so long.
Sklansky points out in an article I'll find later that if a player's variance is plus or minus $500 per day, his variance after n days is not n×500, but √n×500. So, after 100 days, this player's results would be within $5000 of expectations—it wouldn't be off by the $50,000 that simple math might suggest. You can use this data to determine bankroll requirements. (I created a quick spreadsheet using this data, and determined that a player who expected to win $80 a day with a $500 variance would be pretty well guaranteed to be out of the red on the 40th day, and his expected results would never dip below an $800 total loss.)
It occurred to me that this formula can be, well, reformulated. My expectation per hand of $3/$6 is 12¢, plus about $100 and minus about $40. Since I don't know how to rewrite the formula that way, let's say 12¢±$50 per hand. Then this table applies:
| Hands | Min Expect. | Max Expect. |
|---|---|---|
| 10 | –$ 156.91 | $ 159.31 |
| 20 | –$ 221.21 | $ 226.01 |
| 30 | –$ 270.26 | $ 277.46 |
| 40 | –$ 311.43 | $ 321.03 |
| 50 | –$ 347.55 | $ 359.55 |
| 60 | –$ 380.10 | $ 394.50 |
| 70 | –$ 409.93 | $ 426.73 |
| 80 | –$ 437.61 | $ 456.81 |
| 90 | –$ 463.54 | $ 485.14 |
| 100 | –$ 488.00 | $ 512.00 |
| 200 | –$ 683.11 | $ 731.11 |
| 300 | –$ 830.03 | $ 902.03 |
| 400 | –$ 952.00 | $ 1048.00 |
| 500 | –$ 1058.03 | $ 1178.03 |
| 600 | –$ 1152.74 | $ 1296.74 |
| 700 | –$ 1238.88 | $ 1406.88 |
| 800 | –$ 1318.21 | $ 1510.21 |
| 900 | –$ 1392.00 | $ 1608.00 |
| 1000 | –$ 1461.14 | $ 1701.14 |
| 2000 | –$ 1996.07 | $ 2476.07 |
| 3000 | –$ 2378.61 | $ 3098.61 |
| 4000 | –$ 2682.28 | $ 3642.28 |
| 5000 | –$ 2935.53 | $ 4135.53 |
| 6000 | –$ 3152.98 | $ 4592.98 |
| 7000 | –$ 3343.30 | $ 5023.30 |
| 8000 | –$ 3512.14 | $ 5432.14 |
| 9000 | –$ 3663.42 | $ 5823.42 |
| 10,000 | –$ 3800.00 | $ 6200.00 |
| 20,000 | –$ 4671.07 | $ 9471.07 |
| 30,000 | –$ 5060.25 | $ 12,260.25 |
| 40,000 | –$ 5200.00 | $ 14,800.00 |
| 50,000 | –$ 5180.34 | $ 17,180.34 |
| 55,809 | –$ 5114.88 | $ 18,509.04 |
| 60,000 | –$ 5047.45 | $ 19,447.45 |
| 70,000 | –$ 4828.76 | $ 21,628.76 |
| 80,000 | –$ 4542.14 | $ 23,742.14 |
| 90,000 | –$ 4200.00 | $ 25,800.00 |
| 100,000 | –$ 3811.39 | $ 27,811.39 |
| 200,000 | +$ 1639.32 | $ 46,360.68 |
| 300,000 | +$ 8613.87 | $ 63,386.13 |
| 400,000 | +$ 16,377.22 | $ 79,622.78 |
| 500,000 | +$ 24,644.66 | $ 95,355.34 |
| 600,000 | +$ 33,270.17 | $ 110,729.83 |
| 700,000 | +$ 42,167.00 | $ 125,833.00 |
| 800,000 | +$ 51,278.64 | $ 140,721.36 |
| 900,000 | +$ 60,565.84 | $ 155,434.16 |
| 1,000,000 | +$ 70,000.00 | $ 170,000.00 |
| 10,000,000 | $1,041,886.12 | $1,358,113.88 |
My actual number of $3/$6 hands in PokerTracker at this point is 55,809; I put a line for that in my table. My actual $3/$6 results are +$5806.34; this isn't exactly centered within my variance numbers, so probably my guess of ±$50 is off a bit. (Oh, I just looked, and apparently I'm not at 12¢/hand anymore, but a bit over 10¢/hand. Oh well, I don't want to change that table.) But we can still draw some conclusions from this. At around 1000 hands a day, my total for a year should be between 300,000 and 400,000 hands, so as long as I'm not too close to those minimum earnings numbers I should survive just fine as an online pro. Someone with this expectation would have to play 173,612 hands to be guaranteed to be winning, or about six months. The ten million hands numbers I tossed in as a lark; they aren't useful as anything other than a mathematical exercise; that's about thirty years of online play, and whatever online poker is like thirty years from now, my expectation won't be remotely the same.
All of a sudden, I'm not sure what else I can glean from this information. It still seems like the insight was important, but suddenly I'm not sure what to do with it. The answer is clear: Publish this, and let it moulder in the archives of RhymesWithJoker.com.
