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The related GitHub repository with R code.

Previous weekend, I visited the casino with some associates. Of all video games, we get pleasure from North-American-style Baccarat essentially the most. One of these Baccarat is usually referred to as Punto Banco. In brief, Punto Banco is a card sport during which two arms compete: the “participant” and the “banker“. Throughout every coup (a spherical of play), each arms get dealt both 2 or three playing cards, relying on a complicated drawing schema, and all playing cards have a sure worth. Put merely, the hand with the best whole worth of playing cards wins the coup, after which a brand new one begins. Earlier than every coup, gamblers could bet which of the arms will win. Neither hand is in any manner related to the precise home or participant/gambler, so bets could also be positioned on each. All in all, three totally different bets may be positioned in a sport of Punto Banco:

  1. The participant hand has the best whole worth, during which case the participant wins (Punto);
  2. The banker hand has the best whole worth, during which case the banker wins (Banco);
  3. The participant and banker arms have equal whole worth, during which case there’s a tie (Egalité).

If a gambler appropriately bets both Punto or Banco, their bets get a 100% payoff. Nonetheless, a home tax will typically be utilized to Banco wins. As an illustration, Banco wins could solely repay 95% or particular Banco wins (e.g., whole card worth of 5) could repay much less (e.g., 50%). Relying on home guidelines, an accurate bet on a tie (Egalité) will repay both 800% or 900%. A improper bet on Punto or Banco stands in case Egalité is dealt. In all different instances of improper bets, the home takes the cash.

My associates and I like Punto Banco as a result of it’s fully random however appears “gameable”. Punto Banco is performed with six or eight decks so there isn’t any method to know which playing cards can be subsequent. Furthermore, the card-drawing guidelines are fairly complicated, so that you by no means actually know what’s going to occur. Typically each Punto and Banco get solely two playing cards, at different occasions, the hand you bet on will get its third card, which could simply flip issues round. Punto Banco’s perceived gameability comes by means of our human fallacies to see patterns in randomness. Typically, casino’s will place a monitor with the final fifty-so outcomes (see under) to tempt gamblers to (erroneously) spot and bet on patterns. Alternatively, you would possibly suppose it’s sensible to bet in opposition to the desk (play Punto when everyone else goes for Banco) or play on no matter bet received final hand. Because the arms are dealt fairly rapidly in succession, and the minimal bet is usually 10+ euro/greenback, Punto Banco is a fast method to learn the way fortunate you’re.

Image result for punto banco scoreboard
Examples of Baccarat displays, typically positioned subsequent to a desk.

So again to final weekend’s journey to the casino. Sadly, my associates and I misplaced fairly some cash on the Punto Banco desk. We all know the home has an edge (although smaller than in different video games) however usually we’re fairly fortunate. We regularly focus on what can be good methods to reduce this homes’ edge. Clearly, you wish to play as few video games as doable, however that’s so far as we obtained by way of technique. Usually, we simply take a look at our luck and randomly bet Punto or Banco, and sparsely on Egalité.

As a statistical programmer, I assumed it could be attention-grabbing to simulate the sport and its odds from the underside up. On the one hand, I wished to get a way of how favorable the chances are to the home. Then again, I used to be curious as to what extent methods could also be roughly profitable in retaining at the least a few of your hard-earned money.

In my simulations, I observe the Holland On line casino Punto Banco guidelines, which means a six-deck shoe and a Banco win with 5 pays out 50%. I did undertake the extra lenient 9-1 payoff for Egalité although. A number of hours of programming and a few million simulated Baccarat arms later, listed here are the outcomes:

  1. Do not play Baccarat / Punto Banco if you do not need to lose your cash. Clearly, it’s finest to not set foot within the casino in case you can’t afford to lose some cash. Nonetheless, I eagerly pay for the leisure worth I get from it.
  2. You lose least in case you stick with BancoRegardless of having solely a 50% payoff when Banco wins with 5, the chances are finest for Banco because of the drawing guidelines. Certainly, in accordance with the Wizard of Odds, the home edge for Banco (1.06%) is barely decrease than that of Punto (1.24%).
  3. No matter you do, do not bet on Egalité. As a result of most casino’s pay out Eight to 1 in case of a appropriately predicted tie, betting on one appears concerning the worst gambling technique on the market. With a home fringe of over 14%, you’re higher off enjoying most different video games (Wizard of Odds). Though casino’s paying out a tie 9 to 1 lower the home edge to only under 5%, that is nonetheless manner worse than enjoying both Punto or Banco.

The determine under reveals the outcomes of the 5 methods I examined utilizing 50,000 simulations of 100 consecutive arms. Primarily based on the outcomes, I used to be reluctant to develop and take a look at different methods as outcomes look fairly simple: play Banco. Moreover, Wikipedia cites Thorp (1984, unique reference unknown) who instructed that there are not any methods that may actually lead to any important participant benefit, besides perhaps for the endgame of a deck, which presumably requires lots of card counting. For those who however wish to take a look at different methods, please be my visitor, listed here are my 5:

  1. Punto: All the time bet on Punto.
  2. Banco: All the time bet on Banco.
  3. Egalité: All the time bet on Egalité.
  4. LastHand: Guess on the result of the final hand/coup.
  5. LastHand_PB: Guess on the result of the final hand/coup, provided that this was Punto or Banco.


The above determine depicts the anticipated worth of every technique over a sequence of consecutive arms performed. Clearly, the payoff is sort of linear, impartial of your technique. The extra arms you play, the extra you lose. Nonetheless, additionally clear is that some methods outperform others. After 100 arms of Baccarat, enjoying solely Banco will on common lead to a complete loss under the quantity you wager. For instance, in case you bet 10 euro each hand, you should have a lack of about 9 euro’s after 100 rounds, on common. That is in step with the ~1% home edge reported by the Wizard of Odds. Equally, betting solely Punto will lead to a lack of about 130% of the bet quantity, which can be conform the ~1.4% home edge reported by the Wizard of Odds. Betting on Punto or Banco primarily based on whichever received final (LastHand_PB) performs someplace in between these two methods, shedding simply over 100% of the bet quantity in 100 arms. Your anticipated losses enhance while you simply bet on whichever end result got here final, together with Egalité, leading to round ~-150% after 100 arms. That is primarily as a result of betting on Egalité, which appears concerning the worst technique ever, will lead to a exceptional 493.9% loss after 100 arms.

Aside from these common or anticipated values, I used to be additionally within the unfold of outcomes of our hundreds of simulations. Notably as a result of gamblers on a fortunate streak could win far more when betting on Egalité, because the payoff is bigger (8-1 or 9-1). The determine under reveals that any technique together with Egalité will certainly lead to a wider unfold of outcomes. Betting on Egalité could thus be a great technique if you’re by some miracle divinely fortunate, have info on which playing cards are coming subsequent, or have an settlement with the supplier (disclaimer: this can be a joke, please don’t ever bet on Egalité with the intention of being profitable or attempt to cheat on the casino).


If you wish to understand how I programmed these simulations, please go to the related github repository or attain out. I intend on simulating the payoff for numerous different casino video games within the close to future (first up: BlackJack), so if you’re control my web site or twitter.