Any decent simulator makes the number of cards burned and whether they are shown to the players configurable. Although burning one card is pervasive in most casinos, there are some which burn none and others which have various other policies. Most also burn one or more cards when switching dealers.
This may seem insignificant but it effectively reduces penetration by each unseen card burned which can have a measurable effect for a deeply cut game.
I’ve been trying to calculate variance for some time. Let’s use a 6-deck Hit-17 game as an example. Wizard shows:
EV = -0.628%; Variance = 1.346.
The Hi-Lo true count (TC) distribution is:
When -1<TC<+1, 26%;
When +1<=TC<+2, 17%;
When +2<=TC<+3, 9%;
When +3<=TC<+4, 5%;
When +4<=TC, 6%.
If we bet with a spread of $1/$2.5/$5/$7.5/$10 at TC=0/+1/+2/+3/+4 respectfully, the overall variance per hand becomes
[63%x1+(17%x2.5^2+9%x5^2+5%x7.5^2+6%x10^2)]x1.346=17.17.
This calculation assumes one fixed variance component for each TC bucket, but this is not true. So, how does the variance component vary with the TC number?
Just look at any of the hundreds of charts in Chapter 10 of BJA3. Also, see page 20 for the proper calculation methodology. I'm rather surprised that you aren't already familiar with where to find all of this.
Don
I've looked up this part. A typical variance as a function of the true count is like this:
This means that taking insurance brings up the overall variance value. Thank you.
No, you don't know for a fact that that is true and, as it turns out, it isn't! Taking insurance decreases variance. The increase in variance beginning at TCs of +3 is due to more doubling, more splitting, and standing more often on certain hands.
It's dangerous to jump to conclusions.
Don
This is the part I really want to ask. I think it’s still not conclusive on how the insurance affects the variance components, because I haven’t seen the simulation result of variance as a function of true count when insurance is excluded from consideration. I want to see a simulation result of this part using basic strategy without any Hi-Lo deviations. Another concern of mine is the definition of combined variance of these two.
Norm and Gronbog have done all of that. I'm reporting the results.
Don
Some dealers are willing to show the burn card when they switch in. Have noticed a few places that have stopped burning a card when switching dealers. Barona, for many years, would dump the shoe and start a new one when they switch dealers. I don't know if they are still doing this, I haven't been there in a long time.
Dumping the shoe when changing dealers is a red flag for me. It would allow them to implement what is called "preferential shuffling" by changing dealers when the running count is high.
And that would surely be somewhat embarrassing if you had a rash of positive shoes in a row, with the dealers playing tag team. Also, although not recommended, you can always ask to see the burn card and make up some story about superstitious if it's a spade, or an ace or whatever. They'll usually show it to you, but it's doubtful that the information you glean from it is worth the risk of standing out.
Don
You say Kelly is (bank roll x edge)/variance. Let me use this equation on a blackjack game:
At TC=+5, edge=2%, variance=1.264, so Kelly=$10000x2%/1.264=$158;
At TC=+1, edge=0.5%, variance=1.270, so Kelly=$10000x0.5%/1.270=$39.
Here is my question. One table minimum is $10, but the other $25. If we use the same Kelly pattern on these two different table minimums to play all, what are their bet spreads?
The math gets complicated--and there are whole papers written about this--when we impose minimum and/or maximum bets. The ramp is a bit different then, but all the intermediate bets (not min or max) still conform to the same (br x edge)/ variance calculation. It's the unit size that has to be recalculated.
See the Chapter 10 charts for examples.
Don
This one-hand or two-hand play problem has bothered me for some time. The total variance of playing 2 hands at once in a round is
2x(variance + covariance).
If variance=1.346, covariance=0.478, then the total variance of two hands a round is
2x(1.346+0.478)=3.648.
If we use a fixed bet spread of $1/2.5/$5/$7.5/$10 at TC=0/+1/+2/+3/+4 for each of these hand, the total variance per round becomes
[63%x1+(17%x2.5^2+9%x5^2+5%x7.5^2+6%x10^2)]x2x(1.346+0.478)=46.53.
This number does not sound correct because my simulation result shows me that the combined variance is less than two times of individual variance. Could you please consider this?
Your calculations look correct to me, but I'm not understanding what you're asking. What, exactly, do you mean by "less than two times of individual variance"? Are you remembering that, if you're betting optimally, the bet on each of the two hands is only 73% of the one-hand wager?
If you calculate the total variance for betting one hand optimally all the time vs. betting two hands optimally all the time (different unit sizes!), the two-hand variance should be 1.46 times larger than the one-hand variance. Why? Because the EV is also 1.46 larger, and the ratio of EV/variance is what appears in the risk of ruin formula, which is to say that when you bet two hands optimally, compared to one, the EV increases, the variance increases (proportionally), but the ROR remains the same.
Don
I just want to check on this online simulator bjtheorem. With this, I am able to compare one-hand and two-hand EV and SD numbers.
Using a fixed spread of $10/$25/$50/75/$100 for each of these two hands in a round, I got these numbers:
EV=$857; SD=$3030 per 2000 rounds.
However, using a fixed spread of $20/$50/$100/$150/$200 for the only hand in a round, I got these numbers:
EV=$833, SD=$3650, per 2000 rounds.
I thought about this again. Actually, both the simulator and my calculation are correct. They are consistent.
Right. Of course, you aren't betting optimally. If you bet the same amounts per round, the EV will be the same (although if you play alone, you'll win less per hour with two hands), and the variance will be less for two hands.
Don
This variance component problem has bothered me for some time. Consider the insurance side bet only. If we take insurance with one unit of bet whenever it is offered, the expected value (EV) and Variance of Insurance are:
EV = -0.07395; Variance = 1.921.
Now, we use the Hi-Lo system to help take insurance. Betting one unit on insurance at True Count (TC) >=+3 whenever it is offered, what are the EV and variance now?
Page 62 of BJA3, top line may help. Just make the bet one unit instead of 5.26 units, and divide all the subsequent values by 5.26.
Don
I’ve looked up this part. Your number of EV gain from insurance is 0.022% per hand of playing (averaged over all dealer upcards, including Ace and non-Ace). So, roughly speaking, the EV with a dealer upcard of Ace should be 0.022%x13=0.0029. However, the variance number is still missing. Thank you!
How do you take insurance when the dealer's upcard is a non-Ace? And what difference does it make what the variance is? What would you use that number for?
Don
