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The following Table 2 sums the hands probability and actuals based on score, and once again it supports that the simulation is close to what we would expect in theory.The following Table 2 sums the hands probability and actuals based on score, and once again it supports that the simulation is close to what we would expect in theory.

TOTAL Probability Actual Nº Hands
4 0.59% 0.41% 37
5 1.18% 1.32% 119
6 1.78% 1.74% 157
7 2.37% 2.43% 219
8 2.96% 3.00% 270
9 3.55% 3.32% 299
10 4.14% 4.08% 367
11 4.73% 4.72% 425
12 9.47% 9.61% 865
13 9.47% 10.07% 906
14 8.88% 9.14% 823
15 8.28% 7.88% 709
16 7.69% 7.64% 688
17 7.10% 6.68% 601
18 6.51% 6.76% 608
19 5.92% 5.62% 506
20 10.65% 10.74% 967
21 4.73% 4.82% 434

Now, lets look at the hand that the Dealer has been dealt with. The detail is in the following Table 3:

TOTAL Probability Actual Nº Hands
4 0.59% 0.53% 48
5 1.18% 1.21% 109
6 1.78% 1.63% 147
7 2.37% 2.38% 214
8 2.96% 3.01% 271
9 3.55% 3.57% 321
10 4.14% 3.69% 332
11 4.73% 5.02% 452
12 9.47% 9.68% 871
13 9.47% 9.70% 873
14 8.88% 9.13% 822
15 8.28% 8.54% 769
16 7.69% 7.24% 652
17 7.10% 7.38% 664
18 6.51% 6.30% 567
19 5.92% 5.89% 530
20 10.65% 10.07% 906
21 4.73% 5.02% 452


Strategy and Decisions for Playing Blackjack


It seems a simple and easy game and there may not be a method that can always win the Dealer, yet it is quite a difficult task to assess what is the correct decision and reduce the risks. We will look at the following elements in a practical review of 9,000 hands. What is the best number of cards to take? What number should we stick on?

This analysis will help us to evaluate all the decisions we need to make. Split or not to split, double down or not, and if it is possible Surrender. We are using the probabilities of the simulation as it is easier to calculate - effectively the Monte Carlo method. So first, let us look at the various outcomes to see if we have a realistic set of data.

Table 1 has a list of the 169 hands that the Player can be dealt with. Each has the same probability of happening, clearly 1 in 169.

Chart 1 shows that whilst there is a variation, all hands appear between 0.40% and 0.79% with 0.59% as the average.

Again, this would suggest nothing in wrong with our simulation. Now we are ready to analyse the data. The first thing to observe is that the potential outcome cannot be obtained by comparing the above two tables, it merely gives a starting point. The Dealer must go on until at least hitting soft 17, the starting point is obviously an indicator.

All data has been taken from a single deck. I simulated 110 possibilities which can be viewed on the Decision Tree (PDF). It may be excessive, but obviously I wanted to cover as many possible outcomes as possible. It took 3 days to run in Excel!

For my analysis, AA is counted as an initial 12 rather than 2. Not wanting to spoil the surprise but I think the option will be to split this to 11! It should also be noted we pay Blackjack at 3:2. Most of the dealer´s edge can come from the fact that the player acts first, he or she can bust themselves and the dealer does not have to play! It is annoying if you have 13, and the dealer has a 10, you bust with a Queen, and they throw down two 6´s. You lose, it is not a tie! So we are going to start by looking at the dealer´s hand.

Let us look at the 9,000 hands and see what happened with the Dealer´s hand. This can be viewed in Table 4. Incredibly, if we stand every time, the Dealer is forced to go bust on 30% of occasions, reaching the high 40s when opening on a 5 or 6. Even with a high card such as a ten or ace, the Dealer is bust on 1 in 5 hands. The other vital statistic is that a Blackjack only forms 33% of the time from an Ace. Therefore, we can definitely rule out the option of taking Insurance. There is no point. This will save us some of the Blackjack house edge.

Now we know whereabouts the stopping point is for the Dealer. We must work out where we want to stop. For example, on Table 4 there is a 42% chance the Dealer will score over 17 after starting with a 6. If we are on 15, do we want to risk it? What are the chances of our hand busting?

In Table 5 we look at the final score of our simulation and what the previous number was to get them into this position. So, for 21, 100% are busted by taking the next card, 93% are busted from 20. This is to be expected as 7% of A´s are approximately still in there. Still, it isn´t recommended. We then reduce on a sliding scale to 0% of busted from 11 or under. There are a couple of anomalies are 13 to 15, but nothing more than simulation vs. Theory.

In the set of tables in the Table 6 we look at the probabilities of scoring scores after 3, 4 and 5 cards in total. This will help us as certain a stopping position.

Finally, we can now consider the 18 starting scores you could be faced with, and the 10 Dealer scores, and find the "expected" value at each point. This can be summarised in the files of the page Results.

So now lets summarise what we are told from these results. We know that either sticking with 2 cards, going for another, or two more at the maximum gives us our best expected profit. So the first three tables in the group of Table 7 Table 7A , show the expected profit or loss at each decision.

The fourth table now shows us where our Max profit is, and the fifth table, "Number of Cards" informs us where to stop Table 7B And in the Table 7C expected appearances shows the probability of each situation appearing, we apply this and finally, we get a total expected value of Profit of (0.0306)p!!! That is correct, we lose 3p per £ we gamble overall. However, we have several areas that we have not reviewed…"The Extras". We still need to change the probabilities for "Double" Options, "Split" Options and possible "Surrender".

So Table 8 looks at "Doubles". Double down is available under "Reno" rules on 9, 10 or 11. Many European games are only 10 or 11. We will look at "Reno" rules. To make that decision, it is easy, where we show a profit. Let's double. (We have highlighted these already in BJ Results files). For example, for 9, my simulation shows a profit by doubling down on all options from Dealer with a 2, up to and including 7. For 10, double down up to and including 9. For 11, double down up to and including 10, even against the 10 improves us just by 1p.

In Table 9 , we look at Surrender, when we have an expected loss of more than 50p in £, we can reduce this back to 50p by surrendering without playing. We immediately lose 50% of stake. When the Dealer has an Ace, we should surrender with a total of 7, 13 to 16. When the Dealer has a 9 or 10, surrender on 15 and 16.

In Table 10 , we look at the Split options, when do we split or not? One that bucks my thought process was splitting 4's on 8, 9 and 10. Other rules, split 9's when playing a 9…otherwise don't do it. Split all 6, 7, 8 and A's with the exception of 6's against 2's and 7's against 8, 9 and 10, predominantly to minimise losses by 1p and 2p!! Previously, we might have decided to surrender pair of 8's; now we split them. However, a pair of 7's should still be surrendered against an A.

Finally, in Table 11 , we look at totals with A's in. Does considering an A and 6 as a 7 or 17 help? If we consider it as the "hard" number we might make a different decision. It must be stated we are only determining this as to whether to Hit or Not!
Here, we decide that 3, 4 and 5 with an A should be considered "soft", along with 6 and A when faced with a Dealer's 8 or 9. You might have hit anyway. Again, we might have surrendered A with 3, 4 or 5 in previous analysis, now look at it as a soft total.

Now let us put all of these decisions back together in the Table 12A   Table 12B and Table 12C series to see if we can improve our loss of 3p in £. And so what do we get? A loss of £0.0017, this is around 0.2p in the £. It is a result. We always knew overall it would give us a loss; why else would casinos offer the game.