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Blackjack Math

The mathematics of Blackjack have been studied from every possible perspective since 1956, when Roger Baldwin first published an article entitled “Optimum Strategy in Blackjack” in the Journal of the American Statistical Association. One of the foundations of statistical analysis is a function known as “standard deviation,” which plays an important role in understanding the best way to play.

Understanding the Math

The technical definition of standard deviation may seem a bit obtuse for the average Blackjack player, but it is nonetheless the basis for all decision-making during the game. In the simplest terms, it is “a measure of the dispersion of a frequency distribution,” which is represented mathematically as “σ”—the lowercase Greek letter sigma.

Standard deviation is determined by calculation. It is “the square root of the arithmetic mean of the squares of the deviation of each of the class frequencies from the arithmetic mean of the frequency distribution.” In the form of an equation, it can be expressed as

[Add integral here]

where N = the number of values evaluated, x = each value in N, μ = the mean of the values and Σ, of course, is the standard notation for the sum of values from the first value (i = 1) to the last value (N).

The key application of standard deviation to Blackjack is to understand the probability of a net win or a net loss during any given session. Such analysis requires knowing all possible outcomes, which is in turn a function of the rules of the game being played, including how many decks are in play, whether the dealer hits or stands on soft 17, the conditions for splitting and doubling, etc.

Applying the Math

For the sake of example, assume you are playing at a table that uses a six-deck shoe. The dealer stands on soft 17, you may double on any two cards and doubling is allowed after a split. Late surrender is permitted, cards can be re-split up to four hands and Aces may be re-split. In the most extreme case, you could split and double on four hands, resulting in a potential win or loss of eight times (8X) your initial wager.

Without going into great detail, there are 19 possible outcomes for each hand. They include the two extremes (+8X win and -8X loss) as well as a push (0X), a surrender (-0.5X), a simple win or loss (+1X, -1X), a natural blackjack (+1.5X) and all of the various doubling and splitting possibilities. Some of these outcomes obviously have a greater probability of occurrence than others, which is taken into account when making the calculation for the standard deviation with N=19. The end result is σ = 1.1418.

Now how do you use that number? Again, without going into the detailed math, it can be applied to decision making. Calculations show that hitting, standing or surrendering as your first action will most likely result in a net loss of about -4.4% over time. The result of doubling as a first action should yield a 33.8% profit in the long run. And when initial splits are considered, the expected profitability is 14.6%—all based on standard deviation. Obviously, it is to your advantage to double and split whenever possible.

Furthermore, standard deviation can help you manage your bankroll. If you cycle through $1,000 with $10 flat betting (never varying the initial amount wagered), you can calculate that 10% of the time playing a session of 100 hands, you can expect to lose $150. There is a 1% chance you will lose $430.

By contrast, using $1,000 to flat bet $2 a hand in a session of 500 hands, there’s a 10% chance you’ll lose $66, and 1% of the time you can expect a loss of $190. If conserving your bankroll is a consideration, obviously smaller bets over a longer session should be your preference, courtesy of standard deviation.

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