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I once got a call from a fellow who wanted to take $500 to Reno for a weekend, and win a couple of thousand dollars. He asked which book of mine he should buy to accomplish his goal. I told him sorry; but to win a couple of thousand in a weekend requires bigger bets than can be justified with a $500 bankroll.
a) Make bigger bets in those situations where you have an edge, or
b) Find a game with better rules, or
c) Find a game with fewer decks, or
d) Find a game with better penetration, or
e) Play more hands per hour.
Table 82 summarizes the information in most of the rectangles throughout this book. It does not contain double-exposure win rates, and it does not contain win rates for some of the rarer rules. It also contains many win rates that do not appear in rectangles elsewhere in this book.
Every number in table 82 was generated by simula- tion of at least 30 million hands of blackjack. That means the standard error on every number in table 82 is less than $1. In many cases the sample sizes are more than 100 mIllion. Even with samples sizes of 100 mil- lion, differences of $1 are too small to be significant.
Benchmark rules for one, two, and six decks are each based on samples of more than 600 million hands, and each has a standard error of $0.20. Thus the two-deck win rate of $19 is significantly higher than the six-deck win rate of $16. Both are one deck cut off.
You can use table 82 to find win rates for combinations of things not considered as a group in this book. For example, if you are playing blackjack at a casino that allows double after split and late surrender and also has its dealers hit soft seventeen, simply combine the appropriate win rates. There are several ways to do this, and if you have done the arithmetic correctly all methods should come up with the same answer: $20 per hour. One way is to start with the benchmark $16, add $4 for double after splits, add $5 for late surrender, and sub tract $5 for dealers hitting soft seventeen.
The average bet sizes for table 82 are $31.48 for one deck, $27.66 for two decks, and $26.54 for six decks. These are initial bets only, and do not reflect additional amounts wagered on doubles, splits, insurance, etc.
Those little rectangles sprinkled throughout this book have numbers for standard deviation as well as win rate. Those numbers are all around $400. That $400 describes the ups and downs that are typical of an hour of blackjack play with $ 100 maximum bets. Actually, the $400 number is conservative because the benchmark describes making your maximum bet at counts per deck of +4 or more, and such high counts do not happen often. If you are more aggressive on raising your bets, and are willing to go to your maximum ($100 in this example) at a count per deck of +2, then the standard deviation of an hour of play is around $500 instead of $400. lf you go to $100 bets at counts per deck of +1 or more, then the standard deviation applicable to an hour of your play is Over $600. lfyou go to $100 bets at all positive counts, the standard deviation is. $750. If you bet $100 on all non-negative counts, the standard deviation is $800.
These numbers are for playing one hand at a time. If you make the same bet on each of two or more hands. simultaneously, your risk will be higher yet.
The easy way to put it all together is to relate all the numbers to your big bet. Your hourly win rate will be 10% to 50% of your big bet. The standard deviation applicable to an hour of play will be four to eight times your big bet. More on risk later in this chapter.
For example, suppose you have a natural against an ace and you have seen no other cards in a one-deck game. What is the expected value of insurance in this situation? If the dealer has a 10 in the hole, your insurance bet wins double. If the dealer has any other card in the hole, you lose your insurance bet. The probability of the hole card's being a 10 is 15/49, and thus the probability of your winning double the insurance bet is 15/49.
Possible Outcome Probability Product
-1 34/49 -34/49
+2 15/49 30/49
Sum 1 -4/49
The sum of the possible outcomes times probabilities is -4/49. If you
insure all of your naturals, you lose eight cents on every dollar of
insurance. (The above calculation is for single deck, but
multiple-deck calculations yield the same loss of eight cents per
dollar of insurance.) The gambler insures a natural to lock in a
certain winner; you should be looking at the -8% expected value.
This is a simple example of expected value. For most blackjack decisions, calculating the probabilities is more complicated. The definition of expected value is the same - the sum of: possible outcomes times the probabilities of those outcomes.
The same logic applies to other casino games. The dealer's advantage on the pass line at craps is 1.4%; $10 on the pass. line is giving the casino owner 14 cents. Thinking in this manner will take all the fun out of gambling and help turn you into an investor.
Suppose you have a 1.5% advantage on a particular hand of blackjack, and suppose you are betting $50; your expected win on that bet is. seventy-five cents. If you play two hands, in total they are worth a dollar and a half. Your bankroll will go up and down in increments of $50 and $100; but, over the long haul, your average win will be about 75 cents per $50 hand in that situation. You must fight one hand at a time, but the object is to win the war; the war is decided by expected values.
After you have mastered the strategy, a satisfying way to define the battle is by use of a target number of hours played rather than a target number of dollars won. To play the target number of hours is to win the battle.