The Likelihood of Forming a High Hand in Poker

The probability of landing a High Hand with any initial hand combination stands at 63%, which translates to a 2 out of 3 chance.

Definition of a High Hand (High-Card) –

A High Hand is defined as a scenario where players fail to form a pair, thus the strongest card in the hand dictates its strength.

Example – AK372

In Texas Hold'em, High Hands hold the lowest rank where there are neither pairs, draws, nor chances of forming a better hand.

Odds of Making a High Hand on the Flop

It's fairly expected that hitting a High Hand on the flop ranks as the most common occurrence, considering it excludes the potential for Pocket Pairs.

Let’s review some of the odds -

The probability of flopping a high-card hand from any starting hand is recorded at 63%.

The chance of flopping a high-card hand from unpaired starting hands is 66.9%.

When starting with a pocket pair, the odds of flopping a high-card hand drop to 0%.

Although unpaired hands usually flop high-card hands at comparable rates, the frequencies are not identical.

For example -

For an AKo starting hand, the likelihood of flopping a high-card hand is 67.2%.

With T9s as a starting point, the odds of landing a high-card hand is slightly less at 65.4%.

AKo has a marginally better chance of flopping a high-card hand due to its lower frequency of forming straights and flushes compared to T9s.

It’s important to recognize that high-card hands can vary in their level of strength.

The odds of flopping overcards from any unpaired starting hand is calculated at 7.64%.

Odds of flopping overcards with AKo = 67.2%

Odds of flopping overcards with T9s = 16.2%

The chance of flopping a gutshot straight draw from any unpaired starting hand is 10.3%.

Odds of flopping a gutshot with AKo = 11.3%

Odds of flopping a gutshot with T9s = 16.6%

There's a 3.47% probability for flopping an open-ended straight draw (OESD) with any unpaired starting hand.

Odds of flopping an OESD with AKo = 0%

Odds of flopping an OESD with T9s = 9.6%

While a flush draw technically qualifies as a high-card hand, it is significantly stronger than simply flopping naked overcards.

Odds for Later Betting Rounds with a High Hand

We will assume there are no direct draws off the flop but that we hold two strong overcards that could still improve to the best hand should they form a pair.

Overcards grant us 6 viable outs to create the top pair.

Chances of hitting on the turn or river -

The odds of connecting on the turn are 6 out of 47, or approximately 12.8% .

The likelihood of making a hit on the river is 6 out of 46, which translates to around 13% .

It’s crucial to remember that unless our overcards are specifically AK, an overcard could appear on the turn, blocking our chances to achieve the top pair by the river.

Probability of connecting by the river -

To determine the likelihood of hitting by the end of the river, we will calculate our odds of missing and subtract that figure from 100.

Odds of not hitting on the turn = 41/47

Odds of not hitting on the river = 40/46

The chances of failing to hit on either the turn or river are computed as 41/47 multiplied by 40/46, resulting in a 0.7586 probability or roughly 75.9%.

Thus, the probability of hitting by the river is approximately (100 – 75.9), which equals around 24.1% .

However, this 24.1% chance of hitting a pair might present an overly favorable view for several reasons, such as -

- We may hit, yet an overcard could fall by the river, nullifying our chances for a top pair.
- We might hit, but the river could complete a flush.
- We could connect, yet the river might create a straight.

Consequently, pursuing top-pair type holdings is generally not advisable due to the risk of them being outmatched even after striking.

Implied Odds Analysis of a Pair

In the context of Hold’em, one-pair type hands cannot be depended on for their implied odds. Typically, they are regarded as medium-strength hands at best. While a top pair occasionally holds up as the best hand, it often isn't enough to entice significant chip investment. By the time opponents decide to commit a substantial portion of their stack, they will generally have a stronger hand than our top pair.

This dynamic is why one-pair hands are more prone to experiencing reverse implied odds instead of benefiting from implied odds.

Basic Strategy Advice

Hands that consist of a single pair rarely justify risking large stacks (considering a 100bb scenario). When an opponent desires to push their 100bb stack, it's common for them to hold a hand that dominates ours.

Because implied odds are often absent when aiming for a hit, it is generally unwise to attempt to \"chase\" a top pair type hand. Recreational players often enter the fray with Ax type hands preflop, and then find themselves clinging to the hope of hitting an Ace on the flop.

This strategy might make sense if hitting a pair of Aces guaranteed victory, but one-pair hands frequently end up losing in scenarios where more action follows.

Exceptions exist for high-valued TPTK (top-pair-top-kicker) hands, which may sometimes justify committing 100bb, especially when significant chips have already been bet preflop (i.e. in a 3bet or 4bet pot).