The Chances of Achieving a Full House in Poker Games

The likelihood of hitting a full house on the flop when starting with a pocket pair is approximately 0.98%, translating to 1 in 102.

Definition of Full House (also known as a Boat or Full Boat) –

A Full House consists of three cards of one rank and two cards of another rank, meaning you have both three of a kind and a pair simultaneously.

Example – AdAhAsKcKh

A Full House of Aces over Kings is considered the strongest variant of this hand, commonly called 'Aces Full'. poker In the game of poker, being dealt a Full House is quite uncommon, and hitting one on the flop is a rarity.

Odds of Making a Full House on the Flop

The probability of hitting a Full House from any starting hand on the flop is around 0.14%.

If you're playing an unpaired hand, the odds of flopping a Full House drop to approximately 0.09%.

When holding a pocket pair, your chances of flopping a Full House rise to about 0.98%.

You can expect to make a Full House from a pocket pair roughly once in every 100 flops, whereas an unpaired hand will complete a Full House every 1,000 flops. On average, players can anticipate flopping one Full House for every 714 flops they see.

To put that into context;

If we assume that about 25% of hands get to see the flop, the expectation is to flop a Full House approximately once every 2,857 hands.

It's important to note that these calculations represent averages; sometimes you may hit more often, and other times less frequently.

The odds of achieving a Full House or a better hand do not fluctuate significantly, as there aren't many hands that surpass a Full House in strength.

The probability of flopping a Full House or a better hand from any starting hand is 0.17.

For a hand like AKo, the odds of flopping a Full House or better stand at 0.1%.

In the case of T9s, the chance of flopping a Full House or better is around 0.12%.

When holding 88, the odds increase to 1.22% for flopping a Full House or better.

From this, it's clear that a pocket pair is the most favorable scenario for achieving a Full House or better, succeeding nearly ten times more often than unpaired hands.

Chances of Forming a Full House on Later Streets

There are typically two primary scenarios for drawing a Full House after the flop -

where you're aiming for one of your pairs to improve to trips.

1) We have a Two Pair In Scenario 1, it’s straightforward since you know precisely which cards you need to complete your hand.

2) We have Three of a Kind and are hoping that we make a Pair.

Holding two pair, you're looking for a King or a Ten to make a Full House. There are two Kings and two Tens left in the deck, leading to a total of four favorable cards.

Example 1

Flop: KT7
Our Hand: KT

The likelihood of hitting one of these cards on the turn is calculated at 4/47, which equals 0.0851 or approximately 8.5%

On the river, the odds of making that connection increase slightly to 4/46, giving a probability of 0.087 or around 8.7%

Thus, the probability of hitting on either the turn or river

The odds of not hitting on the turn or river total up to 43/47 * 42/46, resulting in a figure of 0.8353 or about 83.5% calculate Odds of not hitting then deduct from 100).

Odds of not hitting on the turn = 43/47

Odds of not hitting on the river = 42/46

Therefore, the chance of hitting is (100 – 83.5), approximately 16.5%

Scenario 2 presents more complexity since you can also form a Full House if the board pairs.

Here, you already hold Three of a Kind and simply need any other card to pair. Either a King or a Seven would give you a total of 6 outs. It's important to note that catching the case Ten (one out) could lead to Quads.

Example 2

Flop KT7
Our Hand: TT

If the turn card is a blank, it still adds another opportunity to hit a pairing by the river. In contrast to the previous example, you will pick up additional outs on the turn if you miss.

Consequently, the odds of hitting on the turn (including the possibility of Quads) rise to 7/47, equating to 0.1489 or about 15%

For the river (also including Quads), the odds improve to 10/46, translating to a probability of 0.2174 or approximately 21.7%

Notably, our chances of making a Full House increase significantly when moving from the turn to the river.

The probability of hitting on either the turn or river can be derived by calculating the odds of not hitting and then subtracting from 100.

The chances of missing on the turn (including the possibility of Quads) stand at 40/47.

The chances of not hitting on the river (considering Quads) is 36/46.

The overall likelihood of missing both the turn and river becomes 40/47 * 36/46 = 0.6660, or approximately 66.6%.

Thus, your chances of hitting by the river are (100-66.6), which comes out to about 33.4%.

As with most strong hands, the value of implied odds varies greatly depending on the strength of your holding. Some Full Houses are stronger than others.