Calculating the Probability of Forming a Flush in Poker

The chance of hitting a Flush on the flop with suited hole cards is 0.82%, which translates to 1 in 122.

Definition of Flush –

A Flush is formed when a player has five cards of the same suit.

Example – AhJh7h3h2h

Among Flushes, the Ace-High is recognized as the strongest variant. Nevertheless, all suits are deemed equal, and only the ranks of the cards determine which Flush prevails over another.

Odds of Making a Flush on the Flop

It's important to note that the likelihood of flopping a Flush is influenced by the type of starting hand dealt to us.

Probability of flopping a Flush with any starting hand is listed as 0.2%.

Probability of flopping a Flush with suited starting hands stands at 0.82%.

Probability of flopping a Flush when holding unsuited cards is effectively 0%.

While the kind of suited hand we hold does not change the odds for forming a Flush, it is true that higher-ranked cards contribute to stronger Flushes. Furthermore, even though the type of suited hand doesn't alter the odds of simply making a Flush, it will impact the odds of achieving a Flush or better.

Probability of flopping a Flush or better with any starting hand is calculated at 0.37%.

Leading odds for making a Flush or better with suited connectors register at 0.94%.

For pocket pairs, the odds of achieving a Flush or better rise to 1.22%.

For an Ace-King offsuit, the probability of achieving a Flush or better is 0.1%.

When holding T9 suited, the chances of making a Flush or better are 0.94%.

Odds of Making a Flush Draw on the Flop

Hitting a Flush on the flop is quite rare and typically requires starting with two suited cards in hand. Yet, drawing for a Flush is more frequent, so it is beneficial to understand the odds of flopping four of the necessary five cards for a Flush.

The odds of flopping a two-card Flush draw with any starting hand amount to 2.58%.

The odds of flopping a Flush draw when holding two suited cards is a notable 10.9%.

When starting with any hand, the odds of flopping a single card nut Flush draw pale at 0.17%.

The odds of achieving a one-card nut Flush draw with an Ax holding boost to 1.12%.

The probability of flopping any one-card Flush draw with an unsuited starting hand stands at 2.24%.

A key takeaway is that we are likely to flop a Flush draw about 11% of the time when we start with a suited hand.

While the likelihood of flopping a one-card Flush draw from an unsuited hand remains fixed at 2.24%, it is critical to remember that non-nut one-card Flush draws hold limited value.

Generally, we prefer our one-card Flush draws to connect to the nuts for better outcomes.

Chances of Achieving a Flush on Later Streets

Let’s suppose we have drawn a Flush on the flop. What are our chances of completing it on the later streets?

Probability of completing a Flush on the turn or river

When we have a Flush draw, there are consistently 9 cards left in the deck that can finalize our Draw.

Hence -

The odds of completing a Flush draw on the turn are calculated as 9/47 = 0.1915, which is approximately 19.2% .

The chances of completing a Flush draw on the river are calculated similarly as 9/46 = 0.1957, approximately 19.6% .

Odds of making a Flush by the river

To simplify, we can compute the probability of not hitting our Flush draw and then deduct that from 100%.

The chance of not hitting a Flush draw on the turn is 38/47.

The chance of not completing a Flush draw on the river is 37/46.

The overall probability of not hitting by the river is calculated as 38/47 * 37/46 = 0.6503 or about 65%.

Thus, the probability of making a Flush by the river becomes (100% - 65%) resulting in roughly 35%.

Implied Odds Analysis of a Flush

The implied odds for making a Flush hinge on two significant factors.

Two-Card versus One-Card Flush – Two-card Flushes (made from both hole cards) typically carry better implied odds compared to one-card Flushes. A two-card Flush on an unpaired board texture is strong enough to engage with 100bb stacks. On the contrary, a non-nut one-card Flush often functions merely as a bluff-catcher in many scenarios.

Paired versus Unpaired Board – Flushes tend to have the highest implied odds when played on unpaired board textures. In contexts with a paired board, Flushes should be approached more cautiously since an opponent might have a Full House already.

Basic Strategy Advice

Two-card strong Flushes are considered highly valuable in Texas Hold'em and are sufficient for playing for 100bb stacks.

Flushes constituted from two cards over paired textures remain robust, although it's generally wise to be cautious about committing all our chips in these situations.

One-card Flushes should only be pursued for stacks when they are the nut Flush. Second-nut one-card Flushes and those of lower rank are considerably weaker in the game.

If an opponent is eager to commit their entire stack, it often indicates they hold a stronger one-card Flush. Thus, mid-strength one-card Flushes typically serve little purpose beyond acting as bluff-catchers in Texas Hold'em.