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How does edge affect your chances of winning at blackjack?

17 October 2011

Conventional wisdom among civilians not inducted into the rites of the casino realm is that the house always wins. The ostensible rationale involves the bosses having an edge in the games they run. The part about having an edge is true – except, of course, in instances like blackjack with card counting where diligent enough players can get an advantage. But as every casino aficionado knows, edge notwithstanding, the establishments don’t inevitably prevail. Solid citizens can and do end sessions, stays, or even extended series of visits profitably. Not because the hucksters let a few folks emerge victorious as bait for the mass of suckers who fail. But owing to the fact that house edge negatively affects earnings but isn’t the only factor influencing gamblers’ fortunes.

Imagine a game in which neither bosses nor bettors enjoy an edge. Single-deck blackjack having rules like the joints offered in bygone days but not any more, played following Basic Strategy, fits the bill. Chance would be 50-50 of realizing a profit or a loss after a few rounds. Assume someone played under such conditions on a table with four spots in action, bankrolled enough to avoid going bust during a run of dismal luck. This person would get about 85 hands per hour. With equal wagers on every round, the probabilities of being ahead or behind by five or more such bets after that hour would be slightly over 31 percent. The corresponding figure for a win or loss of 10 bets or more in the hour would be a bit under 17 percent. After two hours, the figures for being up or down would be 37 and 25 percent at the five- and 10-bet levels, respectively. The determining characteristic is the volatility of the game – the representative bankroll change per hand, measured by what the math mavens call “standard deviation” – equal to 1.13 times the bet.

Were the house to have 0.5 percent edge without altering the other conditions above, chance of finishing anywhere in the black after an hour would be would be 48 percent and in the red the complementary 52 percent. The likelihood of ending five or more bets ahead would be 30 percent and five or more behind 33 percent. For a 10-bet gain or loss in the hour, the likelihoods are 16 or 18 percent, respectively. In two hours, chances are 48 percent of having any profit and 52 percent any loss, 34 percent of savoring the good and 39 percent to the bad by five or more bets, and 23 percent of being ahead and 26 percent behind by 10 or more bets. The wins are possible in statistically small number of trials because volatility swamps edge; house advantage lowers the likelihood a player will be in the lead relative to trailing by the same amount, but not by much.

Competent card counters have a mathematical advantage over the house. The primary means by which they achieve this enviable position is by adjusting bet size in accord with the concentration of high and low cards remaining to be dealt before the next shuffle. High cards help so players raise their bets when the proportion of these not yet drawn exceeds that in a fresh deck, and conversely under the opposite circumstances. Varying bet size raises both the average wager relative to the base amount put at risk on a virgin shoe, and increases standard deviation as well.

Assume, arguendo, that an advantage player gets 1 percent edge by spreading bets with an average equal to 2.25 times the base value and a standard deviation of 3.45. Under these circumstances, the chance of relishing any amount of profit after an hour on a table with four spots in action exceeds 52 percent while that of suffering a loss is close to 47 percent. Being over the top at the end of the hour by at least five base bets has 46 percent probability while suffering a rout of an equal amount has 41 percent probability. At the level of 10 times the base bet for the hour, the chances are 40 percent of rolling in clover and 35 percent of trying to climb out of a hole. The two-hour probabilities of being ahead are 53 percent by any amount, 49 percent by five or more and 44 percent by 10 or more base bets. Conversely, they’re 47 percent of lagging by any amount, 42 percent by five, and 38 percent by 10 or more base bets.

Extraordinarily dedicated blackjack buffs who put in huge amounts of time may reach a point where edge dominates volatility, giving the party with the advantage an exceptionally strong chance of a profit. Consider the prospects after 2,500 hours – about four hours per day, twice a week, for six years averaging four spots in action on the table. Individuals fighting the house’s 0.5 percent edge stand only a 2 percent shot at some level of victory and the complementary 98 percent chance of defeat. Card counters with the 1 percent edge have chances of 99.9 percent of leaving with at least a little of the casino’s dough, and 0.1 percent of its having some of theirs.

So, you can win in the casino when the edge is with the house. Prospects are decent in the short term, especially if your goals aren’t too high relative to the amount you bet. Conversely, card counters can lose; this happens to individuals frequently and is one of the reasons why the most successful of these players operate in teams whose earnings are based on the total number of rounds executed by all of the members. As the poet, Sumner A Ingmark, quaintly quilled:

Uncertainty governs the wide universe,
Some think for the better, and some for the worse,
It may be a blessing, but may be a curse,
Which is it depends on the state of your purse

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.