The probability that I am not just lucky.

I've played about 4,000 hands of what I hope is advantage blackjack. What I would like to determine mathematically is whether I'm actually beating the house, or just lucky.

Typical rules: S17, DAS, resplit any, 3:2 BJ, late surrender, 4-6 deck shoes, 70% penetration
System: Hi-Lo
Spread: first half 1-4, second half 1-8
Wonging: leave at TC -1
Composition-dependent strategy: mostly insurance and H16 v. dealer 10 only, but later on I added H9 v. dealer 2,7 and H8 v. dealer 6.
Tipping: about 0.5 unit bets per hour (yes, I know this kills my EV but I'm out to have fun, not to get rich)

Total, I'm up about 120 units, which actually seems like an absurdly high win rate, and far above what I calculate should be my EV (+0.005).

The problem is that I don't know how to calculate variance. Am I still with the 95% confidence interval of non-advantage blackjack? Am I within the 95% confidence interval for the parameters I've listed?

Your standard deviation is probably around 3 units per hand. If we estimate your EV at 0.01 units per hand (about 1 unit per hour) then your results will look something like this:

EV = 4,000 * 0.01 = 40 units
1 SD = 3 * sqrt(4,000) = 190 units
2 SDs = 380 units
3 SDs = 570 units

That gives us a range of:

1 SD = -150 units to +230 units
2 SDs = -340 units to +420 units
3 SDs = -530 units to +610 units

Your results are about 0.42 standard deviations above your expectation, which is very common (and, luckily, positive ). You would have to play about 90,000 hands to overcome one standard deviation.

-Sonny-

After re-reading your post I see that you estimated your advantage at 0.005 instead of the 0.01 that I used. Adjusting the numbers for that we get:

EV = 4,000 * 0.005 = 20 units
1 SD = 190 units (same)

1 SD = -170 units to +210 units
2 SDs = -360 units to +400 units
3 SDs = -550 units to +590 units

Your results are still within 0.54 standard deviations of your expectation. It will now take you 360,000 hands (about 3,600 hours of play) to overcome one standard deviation.

-Sonny-

Thanks!

May I ask how you eyeballed this?

I used a simulator to get a ballpark figure. The actual results might be slightly smaller since you are using a more timid bet spread at the beginning of the shoe but I don't think they would change much. Most people using a decent bet spread have at least a 2.5-3 unit SD per hand.

-Sonny-

Okey dokey. There's not a chance in hell I'm reaching 90,000 hands anytime soon, so I guess I'll be in the dark for the time being ...

I’m afraid so. With such a small win rate your results will be based primarily on luck for the next 3,6000 hours or so. Your skill won’t really play much of a part. However, if you want to compare you results to the results of someone flat betting and playing basic strategy:

EV = 4,000 * -0.005 = -20 units
SD = sqrt(4,000) * 1.15 = 73 units

1 SD = -93 units to +53 units
2 SDs = -165 units to +125 units
3 SDs = -238 units to +200 units

-Sonny-

You know how I struggle with this stuff all the time lol but I thought maybe the 1% overall adv would be applied to an avg bet unit? Maybe? Could that possibly imply the way you did it that his avg bet unit was only 1 min-unit?