How can I calculate the variance when playing multiple spots but different bet sizes on each of the spots? What is the general formula for n (simultaneous) spots and b(i) different bet sizes on each of the spots?
Thanks,
Takis
Interestingly enough he is the only one who seems to really note that variance and covariance are the only ingredients to these formulas, along with standard deviation, making them formulas for finding what fraction, of what your bankroll allows you to bet on one spot, can be bet on multiple spots. You will find that multiple spots always allow you to be a higher total amount of your bankroll, than the total you can bet on one spot. The question of whether to bet more spots then becomes one of whether multiple spots will cause a premature shuffle by the dealer. Overall follow Wong's rules then.
Maybe you didn't understand my question: I want to bet different amounts on each spot. Say 4 units on the first spot and 6 units on the second spot. What would be my variance at this round?
Wong's formula (and Schlesinger's in BJA 2) assumes equal bet sizes at each of the spots.
But try going over the Wong and Schlesinger explanations of thier formula for betting the same each hand. Now review the slightly different ways they derive this. You should have a clear idea already how to find the variance for a single hand. All you want to drop is the assumption they use, in their final presentations of that formula, that you are betting equal amounts.
That is why I being a bit coy; you should be able to take the final step yourself. I will get back to this thread if you cannot.
In a wacky parallel universe they don't play blackjack but play a game called booger bets. You bet on the flip of a weighted coin, where if you pick your nose you know whether it will land 48/52 heads or 48/52 tails. (One out of 509 people playing booger bets at the Horseshoe, caught picking their nose, get their nose sliced just like Jack Nicholson in the movie Chinatown in the wacky parallel universe described!). Otherwise you are going to just have to guess!
Norm Wattenberger in this universe is the evil twin of the one we know. He gets cranky and tells you that your optimum bet with $5,000 in your playing bankroll is to bet $10 spread over every 2 hands, in a certain casino with certain oportunities to pick your nose. Don in this universe just tells you to read pages 125-126 of Booger Bet Attack. Unfortunately the book binders are as bad in that universe as this one so that page falls out of every copy one week after you buy it.
So gasp you have to depend on the other universe's Clarke Cant to explain what Norm meant. Well math works the same as in the universe here so your overall variance over 2 rounds, for bet A in one round, and B in the other, of 2 is proportional to: A^2+B^2. The minimum variance for bet sizing is when A=B. To compare other ratios of A to B you have varaince proportional in its effect on bet sizing of E= (A^2+B^2)/([(A+B)^2]/2). You have no change when A=B from standard ways to find the optimal bet, and your optimal bet fraction of your bankroll is cut by this amount when you bet different amounts. A must be replaced by A/E, and B by B/E, to get the same risk of ruin etc., as you have when you bet A=B. You can easily show betting 2 times A every other hand and zero otherwise is overbetting, and getting you nowhere.
Now go to the bet sizing formulas for multiple hands given in Professsional Blackjack or Blackjack Attack. Every game reduces down to expected value, variance and covariance when it comes time to optimally size your bets. And what applies to Booger Bets applies to all.
In multiple hand blackjack instead of variance and covariance think of one number being the amount that your variance goes up by playing more than one hand (the way it is used in the usual formulas) and one number being the amount that your variance goes down because one hand(or hands hedge each in general) hedges the other(s) (ditto, same as the parenthetic phrase just given ---clones need ditto defined?).
The good number is divided by E; the bad number multiplied by E.
You should be able to tell which is bad for us and good: variance or covariance?
Now go back to the original formulas, plugging in E for your rather bad camoflage.
It is not a good idea to bet different amounts on multiple hands of blackjack unless your bankroll is so huge you just don't give a damn, or you are playing in a single deck game, with a huge bankroll, and are going for depth charging gains as described in Burning the Tables of Las Vegas.
Now you know why I didn't want to embarass you by telling you directly.
There was no way to be polite in giving this answer and I hope the humor makes up for that rudeness. It should be easy to expand the formulas to show the impact of unequal bet sizes on 3,4 to 7 spots.
