In previous chapters of this series we have applied a number of mathematical methods to improve our Blackjack decision making. Expected final scores, expected earnings and optimal actions. All safe and sound and, yet, we are still not making money at the end of the day. To improve our earnings we will need to do things differently. Consider additional information... but what information could that be?
Until now we have been paying a lot of attention to the information available on our side of the table, the player score. It turns out the information at the other side of the table, the dealer's up card, is also valuable. Factoring in the dealer's up card will help us detect situations where we can achieve higher earnings by introducing exceptions in our strategy.
In chapter 2 we used the dealer strategy to compile a list of all the possible hands the dealer can finish the game with. We then used that list to find out how often we can expect the dealer to end up with a certain score. This information is accurate from a global point of view, but it misses out on a key factor. Knowing the dealer's first card rules out a good number of possible final hands from the list.
| Cards | Score | Probability |
|---|---|---|
| A, A, A, A, A, A, A | 7/17 | 0.00000159% |
| A, A, A, A, A, A, 2 | 8/18 | 0.00000159% |
| … | … | … |
| A, A, K, A, 2, K | Bust | 0.0000207% |
| … | … | … |
| A, 2, 3, 2 | 8/18 | 0.0035% |
| … | … | … |
| 4, 2, 4, 3, 2, 10 | Bust | 0.0000207% |
| … | … | … |
| 8, 9 | 17 | 0.59% |
| … | … | … |
| K, K | 20 | 0.59% |
Source: https://capelski.github.io/blackjack-stats/en/threshold/hands
Take the first final hand in the list, for example: A, A, A, A, A, A, A. This hand is only possible if the dealer's first card is an A. If the dealer's first card is anything but an A, the dealer cannot reach this final hand. We can get more accurate final score probabilities by excluding this final hand from the list when the dealer's first card is not an A.
The same goes for each final hand in the list. If we consider only the final hands that are possible given the dealer's first card, we will get more accurate probabilities. To do so, we need to split the list of final hands into buckets, one for each possible first card. This is what the dealer's final probabilities look like when splitting the list. We can see, for example, that the dealer's probability of getting a blackjack is null when the first card is neither an A nor a 10.
| Dealer score | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 - K | A |
|---|---|---|---|---|---|---|---|---|---|---|
| 17 | 1.08% | 1.04% | 1.00% | 0.94% | 1.27% | 2.84% | 0.99% | 0.92% | 3.43% | 1.01% |
| 18 | 1.04% | 1.00% | 0.97% | 0.94% | 0.82% | 1.06% | 2.76% | 0.92% | 3.43% | 1.01% |
| 19 | 1.00% | 0.97% | 0.93% | 0.91% | 0.82% | 0.60% | 0.99% | 2.70% | 3.43% | 1.01% |
| 20 | 0.95% | 0.93% | 0.90% | 0.87% | 0.78% | 0.60% | 0.53% | 0.92% | 10.53% | 1.01% |
| 21 | 0.91% | 0.88% | 0.86% | 0.83% | 0.75% | 0.57% | 0.53% | 0.47% | 1.06% | 0.41% |
| BJ | - | - | - | - | - | - | - | - | 2.37% | 2.37% |
| 22+ | 2.72% | 2.88% | 3.03% | 3.20% | 3.26% | 2.02% | 1.88% | 1.76% | 6.53% | 0.89% |
| Total | 7.69% | 7.69% | 7.69% | 7.69% | 7.69% | 7.69% | 7.69% | 7.69% | 30.77% | 7.69% |
Source: https://capelski.github.io/blackjack-stats/en/dealer/scores?dcm=absolute
The numbers in the table above reflect the global probabilities. Once the dealer gets their card dealt, we no longer care about the global probabilities. If the dealer's up card is a 2, for example, the probability of the dealer busting is 2.72%, but over the 7.69% total probability for that column, not over the 100% of cases. 2.72% / 7.69% = 35.36%. The dealer will bust 35.36% of the times when their up card is a 2. Let's adjust the probabilities of the table so each column sums up to 100%.
| Dealer score | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 - K | A |
|---|---|---|---|---|---|---|---|---|---|---|
| 17 | 13.98% | 13.50% | 13.05% | 12.23% | 16.54% | 36.86% | 12.86% | 12.00% | 11.14% | 13.08% |
| 18 | 13.49% | 13.05% | 12.59% | 12.23% | 10.63% | 13.78% | 35.93% | 12.00% | 11.14% | 13.08% |
| 19 | 12.97% | 12.56% | 12.14% | 11.77% | 10.63% | 7.86% | 12.86% | 35.08% | 11.14% | 13.08% |
| 20 | 12.40% | 12.03% | 11.65% | 11.31% | 10.17% | 7.86% | 6.94% | 12.00% | 34.22% | 13.08% |
| 21 | 11.80% | 11.47% | 11.12% | 10.82% | 9.72% | 7.41% | 6.94% | 6.08% | 3.45% | 5.39% |
| BJ | - | - | - | - | - | - | - | - | 7.69% | 30.77% |
| 22+ | 35.36% | 37.39% | 39.45% | 41.64% | 42.32% | 26.23% | 24.47% | 22.84% | 21.21% | 11.53% |
Source: https://capelski.github.io/blackjack-stats/en/dealer/scores
The numbers make more sense now. We can now see more clearly that, for example, the probability of the dealer busting is almost four times higher when the dealer card is a 6 compared to when it is an A. Four times higher! This is a huge difference. Surely this must make a difference when computing the edges of the actions.
In chapter 4 we computed the edge of each action for any given player score. Quick recap. The edge of drawing a card comes from pondering the future scenarios we will get to by drawing another card, starting with the highest player score and working our way backwards. The edge of standing comes from comparing the player score with the dealer's expected final scores. In that comparison we used the overall dealer's expected final scores. For a player score of 14, for example, the edge of drawing is -43.68%, while the edge of standing is -42.26%.
Let's see how these edges change when the dealer's up card is a 6, for example. When computing the edge of standing, we will now use the expected final scores for a dealer's 6. The edge of standing improves naturally because the dealer is more likely to bust.
| Dealer score | Probability | Outcome | Edge contribution |
|---|---|---|---|
| 17 | 16.54% | Lose | -16.54% |
| 18 | 10.63% | Lose | -10.63% |
| 19 | 10.63% | Lose | -10.63% |
| 20 | 10.17% | Lose | -10.17% |
| 21 | 9.72% | Lose | -9.72% |
| 22+ | 42.32% | Win | 42.32% |
| Edge | -15.37% |
Source: https://capelski.github.io/blackjack-stats/en/dealer/breakdown/6/analysis/14/?a=stand
The edge of drawing a card changes as well. The optimal action for scores of 15 and higher continues to be the same when facing a dealer 6: standing. Drawing a card therefore leads to the same future scenarios, but each of those scenarios have better edges when facing a dealer 6. Due again to the dealer being more likely to bust. When merging the edges of all the future scenarios, we find that the edge of drawing increases to -30.07%
| Next card | Probability | Next score | Action | Edge | Edge contribution |
|---|---|---|---|---|---|
| A | 1 / 13 | 15 | Stand | -15.37% | -1.18% |
| 2 | 1 / 13 | 16 | Stand | -15.37% | -1.18% |
| 3 | 1 / 13 | 17 | Stand | 1.17% | 0.09% |
| 4 | 1 / 13 | 18 | Stand | 28.34% | 2.18% |
| 5 | 1 / 13 | 19 | Stand | 49.60% | 3.82% |
| 6 | 1 / 13 | 20 | Stand | 70.40% | 5.42% |
| 7 | 1 / 13 | 21 | End | 90.28% | 6.94% |
| 8 - K | 6 / 13 | 22+ | Bust | -100.00% | -46.15% |
| Edge | -30.07% |
Source: https://capelski.github.io/blackjack-stats/en/dealer/breakdown/6/analysis/14/?a=hit
Both edges have increased significantly, but the most important thing is that standing is now much more profitable than drawing a card. We can still expect to lose money, since 14 is a bad player score after all, but, when the dealer has a 6, we will lose less money by standing.
| Action | Global edge | Edge vs dealer 6 |
|---|---|---|
| Stand | -42.26% | -15.37% |
| Hit | -43.68% | -30.07% |
Edges for a player score of 14
A similar change occurs when looking at a player score of 16. Basing the calculation on the overall dealer's expected final scores, the edges of standing and drawing a card are -43.68% and -50.93% respectively. Standing is the optimal action. When the dealer's up card is an A however, the dealer is less likely to bust. The edges are -76.94% and -66.57% when the dealer has an A, and that turns the scale in favour of drawing. You can still expect to lose money when facing an A with a score of 16, but drawing a card will make you lose somewhat less.
| Action | Global edge | Edge vs dealer A |
|---|---|---|
| Stand | -43.68% | -76.94% |
| Hit | -50.93% | -66.57% |
Edges for a player score of 16
This principle applies to all the player scores: the dealer's up card might change the optimal action. Re-computing the optimal actions for each player score and each dealer card yields the following table. Somewhat similar to the optimal actions for the "Stand on 15 and 8/18" strategy, but with a number of exceptions to take advantage of specific dealer weaknesses.
| Player score | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 - K | A |
|---|---|---|---|---|---|---|---|---|---|---|
| 2/12 - 7/17 | H | H | H | H | H | H | H | H | H | H |
| 8/18 | S | S | S | S | S | S | S | H | H | H |
| 9/19 - 10/20 | S | S | S | S | S | S | S | S | S | S |
| 4 - 11 | H | H | H | H | H | H | H | H | H | H |
| 12 | H | H | S | S | S | H | H | H | H | H |
| 13 - 16 | S | S | S | S | S | H | H | H | H | H |
| 17 - 20 | S | S | S | S | S | S | S | S | S | S |
S = Stand / H = Hit. Source: https://capelski.github.io/blackjack-stats/en/dealer/summary?dsm=compact
This is the base of the so called Basic Strategy most Blackjack players use around the globe. At practice, it means having different strategies for different dealer cards. It requires remembering a larger variety of cases, but the increase in complexity is definitely justified by the increase in expected earnings. With these tweaks, we have managed to improve the expected earnings of our strategy to a whopping -2.42%! In the next chapter we will analyze how doubling and splitting in the right moments can help us improve our expected earnings even further.
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