The Odds of Making Four of a Kind in Poker

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The odds of flopping Four of a Kind or better with a pocket pair is 0.24% or 1 in 416

Definition of Four of a Kind (also known as Quads) –

We hold four cards of equal rank.

Example – AdAhAsAcKh

Four of a Kind Aces is the strongest Four of a Kind hand in poker and is also referred to as “Quad Aces”.

Odds of Making a Four of a Kind (Quads) on the Flop

Making Four of a Kind on the flop is an extremely rare occurrence.

We’ll focus solely on the odds of making Four of a Kind or better.

Odds of flopping Four of a Kind or better with any starting hand = 0.03%

Odds of flopping Four of a Kind or better with a pocket pair = 0.24%

Odds of flopping Four of a Kind or better with AKo = 0.01%

To put this into context, assuming any starting hand, we’ll flop Quads or better roughly once every 3,333 flops.

Assuming we play roughly 25% of our starting hands, we can expect to flop Quads or better approximately once every 13,332 hands.

Odds of Making Four of a Kind on the Later Streets

The most common draw to make Four of a Kind is where we already hold Three of a Kind on the current street. Since there are only 4 cards of each rank in the deck, it means that we only have one out to make Quads on each street.

Odds of hitting Quads on the turn = 1/47 = 0.0213 or roughly 2.1%

Odds of hitting Quads on the river = 1/46 = 0.0217 or roughly 2.2%

To calculate the odds of hitting Quads on either the turn or river, we can use a simple trick.

We’ll calculate the probability of not hitting and then subtract from 100.

Odds of not hitting Quads on the turn = 46/47

Odds of not hitting Quads on the river = 45/46

 

Odds of not hitting Quads on either the turn or the river = 46/47 * 45/46 = 0.9574 or roughly 95.7%

 

Hence, the chance of hitting Quads by the river after flopping trips is (100 – 95.7) = roughly 4.3%

So even after making Three of a Kind, we are statistically unlikely to make Four of a Kind by the river.

Implied Odds Analysis of Four of a Kind

Four of a Kind always carries excellent implied odds provided we use at least one of our hole cards.

If the four cards of equal rank are on the board, then every player at the table has Quads, so the implied odds are much less relevant. Any player with an Ace in the hole will win, once there are four cards of the same rank on the board.

Unfortunately, it’s not that likely our opponent will invest a large amount of his stack with the King kicker, so we can’t expect excellent implied odds.

Using one of our hole-cards to make Quads (i.e. three cards of equal rank on the board) carries excellent implied odds. If our opponent has a big pair in the hole, he’ll often assume he has the nuts with his full house and never fold.

He might be aware of the fact that we could hold Quads, but it will rarely be correct for him to make the laydown unless we are playing with very deep effective stacks.

Using two of our hole-cards to make Quads (i.e. two cards of identical rank on the board) is also an extremely valuable configuration in terms of implied odds since our Quads will be relatively disguised and we can frequently “cooler” any full houses our opponent may have.

It does, however, vary by board texture. One issue is that there will be two cards of identical rank on the board, and our opponent may give us credit for having trips even if he never suspects that we might have Quads.

For example, we are holding 55 on 5-5-4-2, and our opponent makes some laydowns, concerned we might hold a 5x. We also know that our opponent can never have a 5x himself due to card removal effects.

Holding 55 on a board texture like 5-5-6-6 is extremely valuable, however, since our opponent will never fold the 6x overfull.

In other words, the exact implied odds of Quads made with a pocket pair depends on the board structure.

Basic Strategy Advice

Quads are basically the nuts. Never fold Quads when 100bb deep (provided we use at least one of our hole cards in formulating the four of a kind).

If the four cards of equal rank are on the board itself, then we should proceed cautiously unless we specifically have the Ace. In such circumstances, we should usually look to continue betting for value even though it’s unlikely we’ll get paid off by worse.

Since this is an uncommon scenario, newer players may make call downs after incorrectly determining the strength of their hand. (For example, they might not realise that their QQ has been counterfeited on the KKKKx board and that any Ace will now win the pot.)         

Odds of making Four of a Kind

 
   

Method (Four of a Kind)

Probability (%)

Flopping Four of a Kind or better with any starting hand

0.03

Flopping Four of a Kind or better with a pocket pair

0.24

Flopping Four of a Kind or better with Ako

0.01

Hitting Quads on the turn with flopped Three of a Kind

2.1

Hitting Quads from turn to river with Three of a Kind

2.2

Hitting Quads by the river with flopped Three of a Kind

4.3

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