When comparing two starting hands, the heads-up probability describes the likelihood of one hand beating the other, assuming that neither hand will fold, and there will be a showdown. This situation occurs quite often in no limit hold'em and tournament play.
An exhaustive analysis of all heads-up match ups in Texas Hold'em poker is made, which required evaluating each possible board for each distinct head-to-head match up, or 169 (number of non-equivalent starting hands out of the 1,326 combinations of choosing 2 cards from 52 cards, assuming that, for example, and are all the same) * 1,225 (number of combinations of choosing 2 cards from a deck of 50 cards) * 1,712,304 (number of combinations of choosing 5 cards from a deck of 48 cards) ≈ 354 billion possibilities. Some techniques are used to avoid redundant calculations.
For example, suppose you are holding pocket aces and you are playing
heads-up against another player holding a random hand.
What is your odds of winning at the showdown if no one folds?!
As illustrated below, in this scenario, you will win 84.93% of the time, lose 14.53% and split the pot 0.54% of the time.
What is the strongest hand against your pocket aces?!
Another pocket pair like pocket kings?! As you can find in the Heads-up Probability Table, pocket kings wins only 17.09% of the time if it has the same suits as your aces and 18.55% of the time if the suits are different from the suits of your aces. It's interesting that pocket kings is even weaker than pocket threes (33) which has 18.68% chance of winning against pocket aces.
Suited connectors?! Yes, but not Jack-Ten though. Surprisingly, ignoring the case when your opponent is also holding pocket aces - which results splitting the pot %95.65 of the time -, the strongest hand against pocket aces is 56 suited (where suits are different from the suits of the aces).
What is the weakest hand against your pocket aces?!
Likewise, it's also counter intuitive that the weakest hand against pocket aces, is A9-off-suit (where 9 has the same suit as one of the aces). Pocket aces wins 93.37% of the time against such hand. AK-off-suit (where king has a suit of one of the aces), is also one of the worst hands against pocket aces. It loses 92.80% of the time against pocket aces.
Give it a try!
See the head-to-head probabilities for your favourite hands here.