Jason1990 asked, "1. Do I have about a 95% chance of winning between -9.2BB and 9.6BB on the next 10 hands I play?" assuming an edge of 0.02 BB per hand and a standard deviation of 1.5 BB per hand.
To answer your question I did some numerical calculations.
First I had to guess at the distribution for one hand. My guess was
70.00% 0 win
20.67% lose 1 big bet
7.06% win 2 to 7 Big Bets
2.27% lose 2 to 6 Big Bets
That distribution has a standard deviation of 1.5 BB and an expectation of 0.02 BB. I also assumed that you always lost or gained an integral number of BB.
I then computed the probabilities for the possible results after playing 10 hands. My results are printed at the bottom of this post.
I found that 94.7261% of the time, we get a result between -9.2 and +9.6 BB. So the answer to question 1 is YES. The Gaussian estimate is accurate even after only 10 hands.
This seems surprising considering that the distribution does not look like a bell shaped curve as noted by others. The Berry-Esseen theorem indicates that the cumulative distribution for 10 hands and the cumulative normal distribution must differ by less than
C rho /(sigma^3 * sqrt(10))
= 0.8* 11.4/(3.37 * 3.16) = 0.85.
When we measure the actual difference, we get differences as large as 0.13. But at the tails of the distribution, the differences are much smaller. So, Berry-Esseen is way too pessimistic for the purposes of these questions.
<font class="small">Code:</font><hr /><pre>
win
or
loss probability
-19 0.000019357419535699934
-18 0.00003600421040582269
-17 0.00006456969332220574
-16 0.00012481787038983035
-15 0.0002612687168103279
-14 0.0005247804087985602
-13 0.0009241500966841833
-12 0.0014654484488216134
-11 0.002395737665248195
-10 0.004454606841964618
-9 0.008541200959866983
-8 0.014395733990861418
-7 0.020100017620170867
-6 0.02570073403965464
-5 0.037722712056824786
-4 0.06518731850759737
-3 0.10496371582704672
-2 0.12977770479990375
-1 0.1114191806112588
0 0.07075359035811012
1 0.05433695876840763
2 0.05865557747163329
3 0.058014682745855146
4 0.05476807202964498
5 0.04626930435645343
6 0.03333247094900252
7 0.021922502380769975
8 0.016717751477048966
9 0.014681501169795198
10 0.01198293096170944
11 0.009280628047952062
12 0.006662885943047578
13 0.004428813219035153
14 0.002913464124472338
15 0.002112229124996278
16 0.0016192430382377574
17 0.0011682355314015158
18 0.0008031858302970064
19 0.0005240124719644412
20 0.0003300763164111946
21 0.0002113263017840343
22 0.00014380247684095702
23 0.00009954016566822779
24 0.00006540745765697863
25 0.00004122060143457475
26 0.000025060547021882077
27 0.00001503408835227676
</pre><hr />