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As late as the mid-1950s, casino operators knew that they made money from their blackjack tables, but none knew exactly what percentage advantage they enjoyed. In his blog, Card Games, Complete With Official Rules (1952), Ely Culbertson pointed out: “In no game that has been played for high stakes has there been less analysis of the science of playing than in Black Jack. The only available guide to strategy is empirical; no one has more than his opinion on which to estimate the advantage of the dealer.” Thankfully, nowadays adhering to the proper BS plays alone is enough to reward players with a slight advantage in single-deck games.
Correct BS is based entirely upon a player’s advantage or disadvantage according to which up-card is showing, i.e., the dealer’s busting potential. There are two ways this can be determined: (1) Baldwin et al. spent three years calculating correct BS plays mathematically according to probability theory- (2) Julian Braun developed computer programs that simulated millions of random hands containing every possible combination of cards, then factored in resulting wins, losses, and pushes to establish the best play in each case. The following chart shows this relationship for every possible dealer up-card. Also shown is the dealer’s chance of busting with the same up-card. From this data it can be calculated that the dealer will bust 28.36% of the time.
From Thorp’s original table, the chart below shows the player’s advantage or disadvantage in percent as specific cards are removed from the deck, if the player follows correct BS.
We can see that the removal of the Aces from a deck results in a financial loss of 2.4% to the player. But surprisingly, when all four 5s are missing, the player has a gain of 3.6%. Therefore, Aces are not as important as one might think, and 5s are more significant than expected. Note too that 8s and 9s are virtually of negligible benefit to either the house or the player.