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Best of Alan Krigman

Gaming Guru

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Why playing multiple spots at blackjack differs from playing one

30 November 2009

Blackjack buffs who wager on multiple spots experience the game differently from solid citizens who bet on single hands. With the same total exposure, the erosive effect the of edge on each round doesn't change. It's the volatility that varies. Technically, volatility is measured by a statistical quantity called "standard deviation." Picture this intuitively as a kind of average by which a bankroll goes up or down as the result of every decision.

Pretend you're flipping coins. Heads wins $2, tails loses $2; neither you nor your opponent have an edge. Your fortune grows or shrinks by $2 per flip. The standard deviation is $2. Instead, say you and a partner play separately but simultaneously, betting $1 each. The same $2 total is up for grabs on every round. One out of the four possible combined outcomes wins $2, one loses $2, and two break even. The average bankroll jump and the standard deviation are $1. Over time, you'll normally be ahead or behind by less than you would with the same number of single $2 flips.

The plot thickens when multiple bets are not independent. Two buddies betting $5 each on nine at the same craps table is just like one person betting $10. Chances are 40 percent to win $14 and 60 percent to lose $10 either way. When each bets $5 on nine at separate tables, however, chances are 16 percent (0.4 x 0.4) to win $14, 48 percent (0.4 x 0.6 + 0.6 x 0.4) to win $2, and 36 percent (0.6 x 0.6) to lose $10 overall. Average bankroll jumps are smaller, owing to the high proportion of $2 outcomes.

In a round of blackjack at a given table, one hand can win while another can lose. So $10 on one spot isn't equivalent to $5 on each of two. But the probability of winning or losing on any round hinges heavily on the dealer's hand; all spots accordingly are subject to the same strength or weakness in this regard.

The standard deviation of a blackjack round per dollar bet on a single hand is $1.13. The excess $0.13 accounts for 3-to-2 payoffs on naturals and coups involving splits and doubles. For one round, per dollar uniformly spread across several hands, the standard deviation is $0.94 for two, $0.87 for three, and $0.83 for four. The formula, in case you think I pulled these values from a hat, is (1/n) x the square root of (1.26 x (n) + 0.5 x (n) x (n-1)) where n is the number of spots, 1.26 is the variance of a single hand, and 0.5 is the covariance relating the hands.

Lower volatility -- smaller average bankroll jumps on every round -- impacts the likelihood you'll go over the top or bite the dust, despite the edge being constant. As an example, make believe you have a $500 bankroll and bet $20 per round. You plan to play until you've doubled your money or lost it all. With one $20 hand, a "risk of ruin" analysis gives the probability you'll succeed as 45 percent. Two hands at $10 each drops this to 43 percent. And four hands at $5 each cuts it further to 41 percent.

Your goal may not be a target profit but a session of at least some desired duration without having to sneak over to the cash machine in the lobby. Two considerations apply. One is, again, the volatility. The other is that the number of rounds you get per hour decreases as more spots are dealt on the table.

Imagine you're the only player. You can figure on 630 rounds with one spot, 420 with two, and 252 with four during three hours. The chance you won't see the lint at bottom of your $500 fanny pack is 58 percent, 78 percent, and 93 percent with $20 on one, $10 on each of two, and $5 on each of four hands, respectively.

Of course, if you normally play one hand at the table minimum, adding spots will increase the total you have at risk. A $10 game doesn't mean $10 on one spot or $5 on two. It'd have to be $10 on each. The cost of edge for a round would rise from $0.05 to $0.10 while standard deviation goes from $10 x 1.13 or $11.30 to $20 x 0.94 or $18.80 for one and two $10 hands respectively. But the casinos have surely taught you that you can't have everything. As the dour doyen of doggerel, Sumner A Ingmark, dutifully declared:

Oh, ye who try to take it all, Who love the big and loathe the small, Prepare thyself to take a fall.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.