Let’s uncover this through a basic statistical assessment!

Blackjack Statistics

This post utilizes a simulation tool for Blackjack constructed with R to derive an optimal approach and the associated likelihoods.

All codes can be accessed on Github.

The regulations implemented for the simulations are:

• Every participant, including the dealer, commences with two cards. One of the dealer’s cards is concealed and will be unveiled at the conclusion of the round, when it’s the dealer’s turn to engage.
• The objective is to request cards to surpass the dealer’s hand without going over 21, each card holding its face value (Kings, Queens, and Jacks are valued at 10). Aces are priced at 1 or 11, whichever value provides the best total without going over. If a hand contains an ace with a value of 11, it is termed as soft, whereas the contrary is referred to as a hard hand.
• If a player surpasses 21 (bust), regardless of the dealer’s score, the dealer secures the wager. Should the dealer exceed 21 and the player not, the player emerges victorious. If both parties possess equivalent scores below 21, the round culminates in a draw. Under all other circumstances, the higher total determines the winner of the round.
• The dealer pays out 1 to 1 for the bet, except for a natural Blackjack (Ace + card worth 10) which pays out at a rate of 3 to 2.
• During each turn, the player can hit (request a card), stand (maintain their current position), or double (the bet is doubled, but only one additional card is drawn).
• In the event that a player possesses two identical cards, they can split, thereby converting their pair into two distinct hands that will be individually played.

The program run.R simulates Blackjack matches (based on 10,000,000 moves with 8 decks and 3 participants) to produce a database that will be scrutinized with the data.table package.

The dataset generated adheres to the following structure, containing a record for each move within a round (game_id) showing the expected earnings should the player hit, stand, or double. The hard_if_hit value denotes if the hand is hard post-move, which is critical for defining the optimal strategy.

All analyses conducted are accessible in the analysis.R script.

Fundamental BlackJack Statistics

The simplest strategy to consider entails standing once the total reaches a specific threshold.

In coding terms, filtering the dataset to extract one entry by game_id that matches the first score exceeding the threshold or the initial score_if_hit if the initial condition is not met suffices.

By implementing this strategy on the dataset for each threshold between 2 and 21, we obtain the subsequent outcomes upon aggregation.

The graph depicted illustrates the anticipated earnings per round (in percentage of the wagered sum) alongside the outcome distribution.

The optimal strategy employs a threshold of 15, resulting in an anticipated loss of 8.57 % of the wagered sum per round. Such odds would inevitably lead to the conclusion of the gameprematurely for the majority of players.

Fortunately, it is viable to formulate a strategy that can enhance these chances by considering the dealer’s total and the presence of soft aces in the player’s hand during the decision phase.

The ultimate BlackJack tactic

In order to identify the supreme strategy, we must initially establish a standard for optimization. We will utilize the forecasted profits following a move.

Calculating this standard for a given hand (total & if the hand is soft) entails understanding:

• the following potential hands when hitting, with the chance of transitioning to each;
• the anticipated profits associated with those hands, based on the tactics we are attempting to establish.

Therefore, we encounter a recursive predicament to address, as we initially have to assess the future probable moves of the game, maintaining the dealer’s total constant. This necessitates implementing a loop that progresses backward on totals. However, the existence of both hard and soft hands necessitates consideration when arranging the sequence.

The following diagram delineates the transitions between hands.

In cases where the player possesses a hard hand with a total exceeding 10, they can only obtain a hard hand with a greater total upon hitting. A soft hand can alter into either a soft hand with a higher total ora hard hand with a total surpassing 10. At last, a hard hand with a total under 9 may convert into a soft hand or a hard hand with a greater total. This implies that our backward loop must incorporate three critical phases.

• Initially, address the hard hands with a total exceeding 10.
• Secondly, handle the soft hands.
• Finally, manage the hard hands with a total less than 9.

This sequential approach guarantees that each phase encompasses all forthcoming possibilities.

For every category, we iterate backward on potential player totals and compute anticipated profits based on weighted profits for all future possibilities (estimated during preceding steps).

The relevant code can be accessed in analysis.R.

Now, we shall delve into the computed strategy for both hard and soft hands.

As predicted, doubling down on totals of 9, 10, or 11 is advantageous due to the heightened likelihood of drawing a card worth 10 (4 out of 13 possibilities). The hitting threshold appears to rise for a dealer’s total exceeding 7, as the dealer’s increased probability of obtaining a 17 and standing necessitates hitting to overcome.

A soft hand of 11 arises solely post-split and corresponds to a single ace. Due to the chance of avoiding busts after a hit, it is evident that the strategy for a soft hand exhibits greater aggression than that for a hard hand.

We can then evaluate scenarios where splitting is optimal by contrasting the projected profits for a total with identical cards and half the total, assuming all else is equal. If the former value is lower than twice the latter (as two hands are wagered post-split), then splitting signifies the move with heightened anticipated profits; if not, we can refer back to the prior diagrams.

The most advantageous split scenarios are as follows:

Splitting aces is consistently advantageous due to the boosted likelihood of obtaining a 10. Conversely, splitting 5s is perpetually discouraged,also due to the chance of getting a 10 or an ace.

The scenario of holding two 9s against a dealer’s 7 is intriguing due to the decision to remain static. This is primarily because the dealer cannot surpass 18 with their next move, forcing them either to stay (and potentially draw) or hit, which carries a high risk of going bust.

The tactics and corresponding profits are accessible via the subsequent gist: https://gist.github.com/ArnaudBu/797094581de3f6703a6c12b994da18c6.

NB: Various strategy charts found online may differ in a few choices. When looking at the anticipated profits in the strategy document above, it is evident that, in such instances, the calculated earnings are quite similar for two separate decisions, indicating a convergence issue that more simulations could resolve.

What is the duration for enjoying a game of Blackjack before running out of funds?

By testing this strategy using simulations (with the test_strategy.R script), one can estimate the average total earnings after a specific number of rounds, plus some quantile ranges.

Thus, the average loss after 1,000 rounds with $1 bets amounts to $8.34. This indicates that the estimated house advantage against this strategy is 0.834%, relatively lower than roulette’s 2.7%.

Assuming an average Blackjack round lasts 1 minute, a player would lose approximately $1.15 after 3 hours with $1 bets each time. The cumulative 5% quantile for loss stands at $26.5. Hence, if one arrives at the casino with 26 times their betting amount, the chances of going broke by the end of a three-hour session are less than 5%.

But what about the option of card counting? With such a minimal house advantage and the significance of 10s in this game, there might be a chance to shift the advantage from the house to the player using this method. Films suggest it’s possible, but does reality align with fiction? This could indeed be an engaging topic for a future post.

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