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How long must you play to hit your blackjack win goal or loss limit?

24 January 2011

You can start a casino game and grab or drop a bundle in blink of an eye. This isn't the usual situation, however. More often, if you size your bets to your bankroll, you'll neither quickly soar to riches nor plunge to insolvency. Instead, you'll enjoy or endure an appropriately long session with the ups and downs that make casino gambling exciting.

The average duration – measured in number of hands – solid citizens experience in a session depends on their exit strategies, bet sizes, and playing tactics. Exit strategies are somewhat artificial because few folks start with more than a nebulous notion of the criteria they'll use to quit, and fewer yet adhere to any intentions they might have had when they began. Still, they're useful models for performing calculations and gaining perspective; this, especially when they're framed in terms of loss limits equal to people's ultimate bankrolls and win goals at which they'd stop without agonizing over whether the streak that put them in the clover would have continued.

To get a handle on the session duration to expect, pretend you play blackjack strictly obeying the dictates of Basic Strategy, at a six-deck table with decent rules such as doubling on any two cards, resplitting, and doubling after splits. These conditions give the house 0.4 percent edge. Further, if you bet the same amount at the start of each round, the volatility of your action has a standard deviation (picture the value as the characteristic bankroll jump per decision) of 1.13 times your wager. A "risk of ruin" calculation using this combination of edge and volatility gives the probabilities of reaching one or the other of your bust-out or profit target exit points. The dens of casino gambling have denizens aplenty – bettors and bosses alike – who think gurus either make all this stuff up or have the patience to develop empirical data gathered by watching huge numbers of games. In fact, those aspects of gambling responsible for moving money from players' pockets to casino coffers follow from fundamental laws of probability. The numbers are not at all occult, but do demand varying degrees of acquaintanceship with arithmetic to ascertain.

To illustrate, here's how the average session duration needed to hit a win goal or loss limit can be determined analytically. Assume you have a bankroll of $500 and are bent on playing until you earn $1,000 or go broke, betting a conservative $10 per round. Your chance of success works out to 23.6 percent and that of failure to the complementary 76.4 percent (OK, formulating the risk of ruin equations and doing the math can get a little complicated so you may have to take my word that these are the probabilities). The amount the casino expects to earn from your action, as a consequence of the edge, is then 76.4 percent of the $500 you leave behind if you lose minus 23.6 percent of the $1,000 you carry away if you win; 0.764 x $500 - 0.236 x $1,000 equals just over $146. Per round, the casino figures the edge gets it 0.4 percent of the $10 you bet, or $0.04. You don't have to be a Nikolai Ivanovitch Lobatchevskii to know that it requires $146 divided by $0.04 or 3,653 rounds for the casino to average $146 at $0.04 a shot.

The accompanying table gives the average numbers of hands corresponding to alternate combinations of win goals and bet sizes for $500 loss limits calculated as above. To lend more intuitive understanding, the table also shows how many hours the indicated number of rounds represent at tables with three and six spots in action (the conversion from rounds per hour as a function of spots in action is, unlike the other factors discussed, based on observation rather than first principles of probability).

Average duration of play in rounds and hours before reaching $500 loss limit or specified win goal for alternate bet sizes

win goal         bet size        rounds  hrs (3 spots)   hrs (6 spots)
$250             $10             1,000   9.5             16.7
$250             $25             158     1.5             2.6
$250             $50             39      0.4             0.6
$500             $10             1,942   18.5            32.4
$500             $25             313     3.0             5.2
$500             $50             78      0.8             1.3
$1,000           $10             3,653   34.8            60.1
$1,000           $25             612     5.8             10.2
$1,000           $50             155     1.5             2.6

The table indicates that the amount of action needed to meet either exit criterion rises sharply as bets decrease, mainly because more decisions become necessary to reach the end-points. There's a secondary effect to keep in mind, too. The more bets players make, smaller though they may be, the greater the gross wagers on which the house edge operates. The casino therefore earns more, lowering the chance players will reach win goals and raising that of hitting loss limits. As an example of the magnitudes involved, for a $500 bankroll and $500 profit objective, prospects of success are 46.9 and 48.43 percent with $25 and $50 bets, respectively. So, what's best? Smaller bets with more expected playing time but less chance of success, or the converse? As is frequently the case in gambling, the pesky personal preference predicament rears its ugly head. The punters' poet, Sumner A Ingmark, put it like this:

Were all decisions cut and dried, a standard plan could be a guide,
But as it is, folks must devise a program based on compromise.

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.