Why Does the Page Say Quads 1 in 4,165 in One Spot and 1 in 4,164 in Another?
You may see four-of-a-kind odds written as “1 in 4,164” in one place and “about 1 in 4,165” in another. Those figures look like they disagree, but they often come from a small wording or rounding difference—not a meaningful change in how rare quads are. The key is to check what is being counted: a five-card hand, a seven-card Texas Hold’em hand, or odds stated “against” rather than “in.”
Here is the example that causes the confusion: Four of a kind odds: 0.0240% (1 in 4,164), and elsewhere the same result may be stated as about 1 in 4,165. Both percentages round to 0.0240%, but the exact five-card calculation gives 1 in 4,165. The number 4,164 is commonly the “odds against” figure. That one-number distinction is small, but it matters when a page labels the number as a probability.
The Short Answer: “1 in” and “Odds Against” Are Different
In a five-card poker hand, there are 2,598,960 possible hands. Of those, 624 are four of a kind. The probability is therefore:
624 ÷ 2,598,960 = 0.0240096%, or 1 in 4,165.
That means one hand out of every 4,165 possible hands is quads, on average. The odds against making quads are different: 2,598,336 non-quad hands for every 624 quad hands, which simplifies to 4,164 to 1 against.
So if a page says “1 in 4,164,” it may have copied the odds-against number into a probability format. Or it may be using a rounded or differently defined calculation. If it is describing the exact probability of a five-card hand, “1 in 4,165” is the precise figure.
Why the Percentage Doesn’t Resolve the Difference
Both 1 in 4,164 and 1 in 4,165 produce a percentage that rounds to 0.0240%. The difference between the exact percentages is tiny: about 0.00000058 percentage points. A display rounded to four decimal places cannot show that distinction.
This is a common source of poker odds rounding confusion. A page may calculate a probability to many decimal places, convert it to a percentage, then round the percentage. Another page may round the denominator, use odds against, or copy a familiar reference figure. The rounded display can look identical even when the underlying format differs.
There is also a practical point: “one in 4,165” does not mean you are due a quad on your 4,165th hand. It describes the long-run average across many hands. You could see quads twice in a short session or go a very long time without seeing them.
Texas Hold’em Changes the Hand-Counting Question
The five-card figure is often quoted as a general poker-hand-ranking statistic. Texas Hold’em, however, gives each player access to seven cards by the river: two hole cards and five community cards. Your best five-card hand is made from any five of those seven. That creates a different probability calculation.
For a complete seven-card Hold’em hand, the probability that the best five-card hand is four of a kind is about 0.168%, or roughly 1 in 595. That is much more frequent than 1 in 4,165 because you are evaluating seven cards and can form the best five-card combination from them.
This does not mean you will make quads in one out of 595 hands dealt at every poker table in exactly that way. The figure is for a random seven-card holding, such as a player’s two cards combined with a fully dealt board, with all possible seven-card combinations counted. If a page instead asks how often a player is dealt pocket pairs, flops a set, or completes quads from a particular starting hand, it is asking a different question.
Check the Event Being Counted
- Five-card hand: The classic 1-in-4,165 probability for quads.
- Seven-card best hand: A different probability, because the player can choose the best five cards from seven.
- Starting-hand event: For example, being dealt a pocket pair or being dealt pocket aces.
- Draw event: For example, starting with a set and making quads by the river.
Before comparing two numbers, make sure they describe the same game, number of cards, and event. A five-card ranking table and a Hold’em draw chart are not interchangeable just because both use the phrase “four of a kind.”
Where Quads Sit in Texas Hold’em Hand Rankings
In standard high poker, four of a kind ranks below a straight flush and above a full house. It consists of four cards of the same rank plus a fifth card, called the kicker. If two players have quads, the higher rank of the four matching cards wins. If those ranks are the same—possible when the quads are on the board—the kicker can decide who wins or whether the pot is split.
Suits do not have different strengths in standard Texas Hold’em. Four kings beat four queens regardless of suit; a spade is not inherently stronger than a heart. When quads are on the board, the best five-card hand may be shared by everyone. Check the kicker: if the board shows four 8s and an ace, players generally use the board’s ace as their kicker and split the pot, unless a player can make a better five-card hand under the exact board conditions.
That last detail is useful in home games, where players sometimes treat their hole card as automatically relevant. It is not. Hold’em hands are the best five-card combination available, and a player may use both hole cards, one, or neither.
Hand Probability Differences: Know Your Denominator
Probability is favorable outcomes divided by all possible outcomes. “Odds against” compares unfavorable outcomes with favorable ones. The two are related, but they are not the same format.
- Identify the hand size. Is the calculation for five cards, seven cards, or a draw from a specific flop?
- Identify the event. Does “quads” mean any five-card quad hand, a seven-card best hand, or improving a set to quads?
- Check the format. Is the figure a percentage, “1 in X” probability, or odds against?
- Check rounding. How many decimal places are shown, and was the denominator rounded?
That process explains many hand probability differences found across poker pages. A chart can be mathematically sound but still be misleading if it leaves out its assumptions. For a trustworthy reference, look for the number of cards, the sample space, and whether the number is a probability or odds against.
Draw Odds by Flop, Turn, and River
Quads draw odds are another calculation entirely. Suppose you hold a pocket pair and the flop gives you three of a kind—a set. There are two cards of your rank left in the deck. If you know your two hole cards and the three flop cards, 47 unseen cards remain. The chance to hit one of those two cards on the turn is 2/47, or about 4.26%.
If you miss the turn, 46 wheel straight poker meaning unseen cards remain for the river, so the chance of hitting on the river specifically is 2/46, or about 4.35%. The chance to hit at least one of the two remaining cards by the river is:
1 − (45/47 × 44/46) = about 8.42%.
That is a useful example of why “by the river” is not the same as doubling the turn percentage exactly. The deck gets smaller after high card odds texas holdem the turn, and the calculation accounts for both streets. It also assumes you will see both cards; in a real hand, betting may make you fold or charge you to continue.
Do not confuse this with a flush draw. A flush draw is about completing five cards of the same suit; a set-to-quads draw is about finding one of two remaining cards of your rank. They are different outs and different calculations. Suits do not have unequal value, and holding a spade does not improve a quads draw unless it changes the actual five-card hand.
How Printable Cheat Sheets and Equity Charts Help
Printable cheat sheets are handy for learning hand rankings, basic outs, and common draw percentages without stopping a game to search online. A good sheet should label whether its odds are for one card to come or two cards to come, and whether it assumes you see the river for free. If it lists quads as 1 in 4,164, look for a note saying “odds against”; if it calls that a probability, the wording may be imprecise.
Equity charts answer a different question. They estimate how often a hand wins or ties against another hand or range, given the cards known and the cards still to come. For example, a pocket pair’s chance of improving to quads is not the same as its equity against an opponent’s range. You can miss your quads draw and still win with a full house or a lower set; conversely, you can make quads and still need to consider whether the board allows a straight flush.
Use a chart as a learning aid, not as a substitute for checking the situation. If villain has a higher set and you have a lower set, your two remaining rank cards may be outs—but the board texture and possible redraws matter. If the board pairs, villain could already have a full house, so count the actual winning cards rather than assuming every improvement wins.
What to Do When a Poker Page Lists Both Numbers
If you see “Four of a kind odds: 0.0240% (1 in 4,164)” and another section says “about 1 in 4,165,” the most likely explanation is that one figure uses the odds-against denominator while the other gives the probability denominator. The rounded percentage conceals the difference. For the exact five-card calculation, 0.0240096% is 1 in 4,165; 4,164 to 1 is the corresponding odds against.
If the page is about Texas Hold’em, also check whether it is counting five-card hands, seven-card best hands, or a specific draw from a flop. Those are not the same statistic, and a chart should say which one it means.
The practical takeaway is simple: read the label, not just the number. “1 in X,” “X to 1 against,” and a rounded percentage each communicate different things. Once you know the hand size and the event being counted, the 4,164-versus-4,165 puzzle stops being a poker mystery—and you can focus on the more important decision at the table: what hands your opponent can actually have.