# First Two Cards & Dealer Upcard Frequencies

### What Blackjack hands do we get?

On the previous pages we looked at the 34 possible initial hands that the player can get. But the dealer upcard is nearly as important. Here we will look a bit deeper by including the dealer upcard. As there are ten possible upcards, there are 340 possible initial two-card hands and dealer upcard. These charts are more complex and require three dimensions. As in the previous charts, the x-axis shows each of the 34 initial two card hands. The third dimension (depth) shows the ten possible dealer upcards. The peaks show the highest frequency hands. The first chart on this page is the simplest three-d chart. The most common hand is T,T vs. T. This is the very sharp peak at the far right-rear. The next most common hands are player 12 and 13. The hands with a dealer upcard of Ten are about four times as likely as the hands with upcards of ace–nine because there are four ties as many Tens in the shoe. However, they are not exactly four times as common since a dealer upcard of Ten means there is slightly less likelihood of your hand having a Ten. The legend at the lower left defines the colors. Red indicates that a hand occurs less than 0.5% of the time. Green indicates 0.5% to 1%, etc.

### What are the good Blackjack hands?

Now it gets more complicated. Here we see the same 340 hands. But instead of the frequency of each hand, we see the advantage of each hand. That is, average gain or loss if you get the hand. Green and Red have a negative advantage. Blue, Yellow and Purple have increasing degrees of positive advantage. Obviously Blackjack is the best hand. You can see a gain of 150% except against an Ace or Ten where there is a bit of a downward slope. As you would expect, stiffs (12-16) are the worst hands and 17 is nearly as bad. But this chart also shows that the upcard is very important. You can see red areas on either side of the chart where the advantage falls below -50%. These are hands against a dealer Ten or Ace. And you will see a ridge down the center of the chart where the dealer upcard is a six. A dealer six is particularly good with player pairs. It is no surprise that 16 against Ten and Ace are the lowest valleys in the chart.

### What Blackjack hands get the money?

First we looked at how often we get each hand. Then at the advantage of each hand. Now we look at the overall win rate or gain from each hand. What's the difference between advantage and win rate? Well a hand may have a high advantage when we get it. But if we don't get it very often, its overall value is less. Win rate is advantage times frequency and tells us the overall money won or lost for each possible hand. This chart looks quite a bit different. Here blue indicates a loss and the other colors indicate increasing wins. Blackjacks are still the most valuable hands. Blackjack against a Ten is the largest gainer. It's a scary hand since the dealer may also have a BJ. But, it is more common and therefore results in a higher total gain. Pairs and soft hands now show a very small gain or loss due to their low frequency. Ten, Ten is very important not only because it is a good hand but because it occurs so frequently. The Advantage chart showed 16 to be the worst hand. Not surprising as you will probably lose when you get it. But this chart shows 13 and 15 as the worst hands because you will still probably lose them, but they occur a bit more often. As in the previous chart, the gain against a dealer Ten or Ace is lower and in some cases there is a loss against these cards where a gain existed against other dealer upcards.

### Sim details

• Six decks, S17, DAS, LS, Heads-Up, Basic Strategy, 75% Penetration
• Ten billion rounds each