Let’s uncover this using a straightforward statistical evaluation!

Blackjack Statistics

This post utilizes a simulation program for BlackJack created with R to derive an optimized approach and the pertinent probabilities.

All the scripts can be accessed on Github.

The regulations followed for the simulator include:

• Every participant, including the dealer, commences with two cards. One card belonging to the dealer remains concealed and only becomes visible at the end of the round, during the dealer’s turn.
• The objective is to request cards to surpass the dealer’s hand without going over 21, with each card contributing its face value (Tens, Queens, and Kings have a value of 10). Aces are valued at 1 or 11, whichever value grants the best score without exceeding 21. A hand containing an Ace with a value of 11 is labeled as soft, while the opposite is termed a hard hand.
• If a player exceeds 21 (busts), regardless of the dealer’s total, the dealer clinches the wager. If the dealer busts while the player doesn’t, the player wins. In situations with matching scores below 21, the round concludes in a tie. In all other instances, the higher score emerges victorious for the round.
• The dealer pays out the wager at 1 to 1, unless it’s a natural Blackjack (Ace + card with a value of 10), which pays 3 to 2.
• During each player’s turn, they may hit (ask for a card), stand (stay at the current position), or double (the bet is doubled but only one additional card can be drawn).
• If a player possesses two identical cards, they have the option to split, dividing their pair into two distinct hands that will be independently played.

The program run.R runs simulations of BlackJack games (based on 10,000,000 moves with 8 decks and 3 players participating) to create a dataset that will be assessed using the data.table package.

The resultant dataset adheres to the layout below, where each line represents a move in a round (game_id) with the projected earnings if the player decides to hit, stand, or double. The hard_if_hit value indicates whether the hand becomes hard after the move, crucial for defining the optimal strategy.

All the analyses conducted can be reviewed in the analysis.R script.

Key BlackJack Details

A fundamental strategy involves standing once the score surpasses a specific threshold.

In coding terms, we simply filter the dataset to extract a single entry by game_id that matches the initial score that goes beyond the threshold, or the first score_if_hit that does so if the initial scenario is not satisfied.

By applying this approach to the dataset for each threshold from 2 to 21, we compile the following outcomes after consolidation.

The chart above illustrates the anticipated earnings for each round (as a percentage of the wagered sum) along with the outcome distribution.

For an optimal strategy, a threshold of 15 is favored, resulting in an anticipated loss of 8.57% of the stake per round.

These chances would evidently prompt the game to terminate prematurely for most participants.

Luckily, a strategy can be formulated to enhance these probabilities by considering the dealer’s total and the soft aces in the player’s possession during the decision-making procedure.

The most effective BlackJack tactic

To uncover the best strategy, it is essential to establish a criterion for optimization. The metric employed is the anticipated profits following a move.

Evaluating this measure for a particular hand (total & hand softness) requires awareness of:

• the potential subsequent hands from hitting, along with the chances of transitioning to each;
• the anticipated profits linked to those hands, based on the actions stipulated by the impending strategy.

Consequently, a recursive predicament arises where the forthcoming moves in the round must be gauged, assuming the dealer’s total remains constant. This necessitates instating a loop that traverses backwards through scores. However, the existence of both hard and soft hands compels us to consider this aspect while organizing our steps.

The ensuing diagram delineates the hand transitions.

In case a player holds a firm hand with a total surpassing 10, only a firmer hand with a higher total results from hitting. A soft hand can convert to either a softer one with a superior total or a stiffer hand surpassing 10. Lastly, a rigid hand with a total less than 9 can create a soft hand or a firmer hand with a higher total. Hence, we need to sequence our reverse loop into three pivotal stages.

• Initiate with firm hands scoring above 10.
• Proceed with soft hands next.
• Conclude with firm hands totaling less than 9.

This arrangement guarantees each future possibility is duly addressed at every juncture.

For each class, we move backwards across conceivable player totals and calculate anticipated profits predicated on the weighted profits for all forthcoming possibilities (assessed in prior stages).

The script can be accessed in analysis.R.

Let’s now delve into the devised strategy for both firm and soft hands.

Figure 3 — Decision blueprint for solid hands

It is advisable to double down on totals of 9, 10, or 11 owing to the heightened likelihood of drawing a 10-valued card (4 out of 13 chances). The threshold for hitting appears to rise when the dealer’s total exceeds 7, due to the increased likelihood of the dealer achieving a total of 17 and standing, necessitating a hit to surpass them.

Analogous to a single ace, a soft total of 11 can only be achieved post-splitting. The strategy for soft hands is notably more assertive compared to that for solid hands, attributable to the chance of not exceeding the limit post-hit.

One can deduce the scenarios where splitting is most advantageous by contrasting the anticipated profits for a total with identical cards against the earnings for half that total, all else equal. If the former value is lower than twice the latter (betting on two hands post-split), then splitting emerges as the more profitable move; otherwise, reference can be made to the prior visuals.

Hence, the optimal split scenarios are:

Splitting aces always emerges as a compelling choice, thanks to the heightened likelihood of attaining a 10.Contrarily, dividing 5s is not advisable, primarily due to the chance of getting a 10 or an ace.

The scenario involving a hand of two 9s versus a dealer’s total of 7 presents an intriguing situation where opting to stay appears distinctive. This is once again due to the dealer’s inability to surpass 18 on the subsequent move, forcing them into a position where they must stay (and draw in the best-case scenario) or opt to hit, leading to a high risk of going bust.

The tactics and the correlated profits can be accessed via the ensuing link: https://gist.github.com/ArnaudBu/797094581de3f6703a6c12b994da18c6.

NB: strategy charts found online may differ on certain decisions. By reviewing the expected profits from the strategy document above, we observe that in these instances, the calculated profits for two distinct decisions are very close, indicating a convergence issue that can be addressed by running more simulations.

What is the duration of entertainment in a Blackjack game before encountering bankruptcy?

By testing this strategy against simulations (using the test_strategy.R script), we can approximate the average earnings accumulated after a specific number of rounds, along with diverse quantile ranges.

Hence, after 1,000 rounds, the average deficit amounts to $8.34 for $1 bets. This implies that the assessed casino advantage against this strategy is 0.834%, a comparative low against the 2.7% of roulette.

Considering the average duration of a Blackjack round as 1 minute, on average, the player will lose $1.15 after 3 hours, wagering $1 at each instance. The cumulative 5% quantile for losses stands at $26.5. Therefore, if you enter the casino with 26 times your wagering amount, there is less than a 5% probability of experiencing bankruptcy by the conclusion of your three-hour session.

What about the feasibility of card counting? With such a marginal house advantage and the significance of 10s in this game, it might be possible to shift the advantage from the casino to the player using this tactic. Movies affirm it, but does reality align with the depiction? This certainly presents an engaging subject for a forthcoming post.

Stats were also taken from these US cities and states:

CITIES: