This grid compares the strength of a particular Texas hold'em hand (in this case ) against all other hands. Each square represents a pair of opponent cards, and the color represents the relative strength of these two hands — the redder the square, the more likely the opponent hand is to win. These probabilities were computed by enumerating all ≈1.3 trillion hands of heads-up Texas hold'em.

Opponent pair cards are organized along the main diagonal, in a band of 4x4 boxes. Opponent suited cards lie along other diagonals. The dark squares correspond to impossible hands, where a card is duplicated.

If you are holding a pair, an opponent holding any higher pair has essentially the same advantage over you, regardless of the rank of the pair. In fact, compared to (say) a pair of 6s, a pair of 7s is slightly better than a pair of Aces.

While a pair of Aces is the strongest hand averaged over all opponent hands, the single strongest heads-up face-off is a pair of Kings vs a K+2 offsuit.

You can think of the full dataset of probabilities as a 4D hypercube (one dimension for each of the four cards dealt between two players). The panel above is a 2D slice through this 4D cube; it is a detailed view of the strength of one particular hand.

The average color in the above panel is a single number summarizing the strength of a hand averaged over all opponent hands. Computing this for all hand combinations produces the panel on the left. If you like thinking about the first panel as a 2D slice through a 4D cube, then this second panel is a projection of the 4D cube into 2D.

Every square tinted blue corresponds to a hand that wins more often than it loses, when playing against a random hand. However, it would be unwise to bet on the weaker blue hands (e.g., Q♣4♣). Why? Our procedure for averaging each hand over all opponent hands assumes that every opponent hand is equally likely. Since opponents can fold weak hands, that isn't true.

We can compute a more accurate summary of hand strength if we know how often real opponents tend to play every hand. Luckily, such databases exist. This third panel shows the actual frequency with which each hand is played, based on mining several million online poker hands. The brightest squares are played most often.

As we would expect, this panel correlates well with the panel above — real opponents are more likely to play stronger hands. There are some surprises as well. For example, see the brighter box in the upper-right corner of the panel, beginning at rank 10? This jump in hand frequency is more pronounced than the equivalent increase in hand strength. Maybe this indicates a psycological bias that overvalues (and overplays) facecards.

With the hand frequency data above, we can compute a more accurate average hand strength. We repeat the same process for averaging each hand over all opponent hands, but we weight the average by the frequency of each opponent hand. The colors in this panel correspond to the profitiability of each hand, when playing against a "typical" opponent who folds weak hands.