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# How Pushes Affect House Advantage in Casino Games

6 July 2005

Some casino bets either win or lose. Others have a third possibility, they can "push," neither winning nor losing. In games where pushes can occur, they may play a more significant role than most folks realize. Always, although superficially, because the action is slower with fewer exchanges of money in any given number of rounds or time period. Occasionally, yet more substantively, when pushes change the house advantage by shaving the probabilities of losing and winning by different amounts.

To illustrate this phenomenon, envision a hypothetical game with outcomes dependent on players' decisions. Assume a particular even-money proposition can be handled in two ways. One offers 46 percent chance of winning and 54 percent of losing. The other has 45 percent chance of winning, 5 percent of pushing, and 50 percent of losing. Which is preferable?

Based only on likelihood of winning, the first modus operandi would be better than the second, 46 versus 45 percent. Considering just prospects of losing, the second would outshine the first, 50 rather than 53 percent. House advantage or edge combines both factors, accounting for relative shifts in expected losses and wins. With even-money wagers, edge equals the chance of losing minus that of winning. In the respective cases cited, it's (53 - 46) = 7 percent and (50 - 45) = 5 percent. The edge in this example is less with the decision that allows for pushes, suggesting it as the superior long-term approach.

The enigmatic blackjack decision hit or stand on a 12 against a dealer's two-up shows the potential pertinence of pushes in the casino simulacrum of reality. And, for solid citizens whose quest for enlightenment goes past assuming "the unseen card is a 10," helps explain why "you're supposed to" hit this hand.

Blackjack buffs who stand on 12 win only if the dealer busts and can't push. The chance of the dealer busting and therefore of players winning when they stand is 35.36 percent. The probability of the dealer finishing from 17 through 21 and beating players who stand at 12 is the complementary 64.64 percent. The joint's juice is accordingly (64.64 - 35.36) = 29.28 percent.

Players who hit may exceed 21 and fall then and there, or reach totals from 13 to 21. The chance of an immediate bust is 30.77 percent. Likewise for totals under 17, where the hand hasn't improved and loses unless the dealer breaks. Each of the totals from 17 through 21 has a 7.69 percent chance of being attained. A 17 can push but can't win unless the dealer crashes and burns; the others push if the dealer matches them; they win if the dealer busts or ends with a lower total. All told, when players hit 12 against two-up, they're projected to win 34.82 percent of their hands, push 4.97 percent, and lose 60.21 percent. The house advantage is therefore (60.21 - 34.82) = 25.39 percent.

Theoretical outcomes for hitting and standing on 12 versus two-up are summarized in the accompanying table. The figures show this particular hand to be an underdog regardless of how executed.

Theoretical outcomes of alternate ways to
handle 12 versus dealer two-up

 Result Hit Once Stand Player wins 34.82% 35.36% Push 4.97% 0.00% Player loses 60.21% 64.64% House edge 25.39% 29.28%

Under some circumstances, an argument can be made for picking the option predicted to win most often. Standing might get the nod in this instance, but the improvement over hitting is a scant 0.56 percent. The decrease in the chance of losing when players hit rather than stand is much greater, at 4.43 percent. And, the edge represents 3.89 percent less penalty hitting than standing. All because of pushes you might not have thought important. Since basic strategy in blackjack comprises the moves giving the bosses the least edge on adverse hands or the players the most under favorable conditions, "the book" says to hit. Recalling the rhyming rejoinder of the Punter's Poet, Sumner A Ingmark:

In settling on the tactical technique you're gonna use,
Remember that not winning doesn't always mean you lose.n

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Best of Alan Krigman
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.