Poker Hand Calculator

A poker hand calculator computes the probability of being dealt any specific hand in five-card poker, compares the strength of two or more hands, and calculates the odds of improving a hand on future cards. It is used by poker players studying probability, maths students working on combinatorics, and anyone curious about the mathematics behind one of the world's most popular card games.

How to Use the Poker Hand Calculator

1. Select your calculation type: hand probability, hand comparison, or draw odds.
2. For hand probability, choose the hand type (royal flush, full house, two pair, etc.) to see the exact probability and frequency.
3. For hand comparison, enter two or more hands using standard card notation (e.g. Ah Kh Qh Jh Th) to see which wins and the percentage chance of each hand winning.
4. For draw odds, enter your current hand and remaining community cards to see your outs and probability of improvement.
5. Review the pot odds comparison to see whether a call is mathematically justified.

The Formula

Total number of 5-card hands from a 52-card deck: C(52, 5) = 2,598,960

Probability of a hand = number of ways to make that hand / 2,598,960

The number of ways to make each hand is calculated using combinatorics.

Royal flush: 4 ways (one per suit). Probability = 4/2,598,960 = 0.000154%

Full house: C(13,1) x C(4,3) x C(12,1) x C(4,2) = 13 x 4 x 12 x 6 = 3,744 ways.

For draw odds: outs / remaining cards. With two cards to come, use the rule of four (multiply outs by 4 for an approximation). With one card to come, use the rule of two.

Real-World Example

You hold two hearts in your hand and there are two hearts on the flop in Texas Hold'em. How likely are you to complete the flush by the river?

Outs: 13 hearts in the deck, 4 already visible = 9 remaining hearts. Total unseen cards: 52 - 5 (your hand + flop) = 47 cards remaining.

After the turn (one card to come): probability = 9/46 = 19.6% After the flop (two cards to come): 1 - (38/47 x 37/46) = 1 - 0.651 = 34.9%

Rule of four shortcut: 9 outs x 4 = 36% (close approximation to 34.9%). Rule of two shortcut: 9 outs x 2 = 18% (close to 19.6% after the turn).

Poker Odds and Expected Value

Understanding probability is fundamental to long-term success in poker. The concept of pot odds connects the probability of making your hand to the profitability of calling a bet. If the pot contains £100 and your opponent bets £50, you must call £50 to win £150, giving pot odds of 3:1. If your probability of winning is greater than 1 in 4 (25%), the call has positive expected value. Over thousands of hands, making decisions with positive expected value leads to profit. The reverse is also true: folding when pot odds are favourable is a long-run mistake. Implied odds extend this calculation by accounting for future bets you expect to win if you hit your hand. Poker mathematics also includes the study of hand ranges (the distribution of hands an opponent might hold) and equity calculation (your share of the pot given all possible outcomes).

Frequently Asked Questions

What is the rarest hand in poker? The royal flush is the rarest and highest-ranking hand: the ace, king, queen, jack, and ten of the same suit. There are exactly four royal flushes (one per suit) in a standard 52-card deck, giving a probability of approximately 1 in 649,740, or 0.000154%. On average, you would need to play well over half a million hands to be dealt one in a five-card draw game.

What are "outs" in poker? Outs are the cards remaining in the deck that will improve your hand. For example, if you have four cards to a flush, there are 9 remaining cards of your suit in the deck (13 total minus the 4 you can see), so you have 9 outs. If you have an open-ended straight draw (needing either end to complete it), you typically have 8 outs. The more outs you have, the more likely you are to improve.

What is the difference between odds and probability? Probability is expressed as a fraction or percentage (9/47 = 19.1%). Odds express the same information as a ratio of failure to success (38:9, or approximately 4.2:1). Poker players often use odds because pot odds are also expressed as ratios, making comparisons straightforward. If your odds of hitting your hand are 4:1 and the pot is offering 5:1, you have the correct odds to call.

Does hand ranking change in different poker variants? The standard hand rankings (royal flush, straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, one pair, high card) apply in most variants including Texas Hold'em, Omaha, and Five-Card Draw. Some variants introduce additional hands (such as five of a kind when wild cards are used) or alter the ranking (in Lowball games, the lowest hand wins). The calculator covers standard hand rankings by default.

Every five-card hand in one table

The page gives the total number of five-card hands as C(52, 5) = 2,598,960 and works the full house out in full. The remaining hand types sit in the same denominator, and the ten counts below add to that total exactly.

The royal flush row reproduces the page's 4 ways and 0.000154 percent. The full house row reproduces its 3,744. The counts add to 2,598,960, which is the first check on the whole table: any hand type missing a combination that belongs to it shows up as a total that misses the denominator.

Why the flush count is 5,108 and not 5,148

Reading a flush as five cards of one suit in any order gives 4 x C(13, 5) = 4 x 1,287 = 5,148. That figure is too high, because a straight flush is also five cards of one suit and it has already been counted as a straight flush.

The same subtraction governs the straight. Ten rank sequences are possible, from ace to five up to ten to ace, and each card in a sequence may be any of four suits, which gives 10 x 4^5 = 10,240. The 40 straight flushes are excluded, and 10,240 minus 40 leaves 10,200. Both counts are the ones in the table above.

High card is the last count to fall out, and it is the only one that excludes sequences rather than suits: (C(13, 5) minus 10) x (4^5 minus 4), which is 1,277 x 1,020 = 1,302,540.

Outs against exact odds

The page gives the flush draw as 9 outs, 19.6 percent with one card to come and 34.9 percent with two. Extending the same working across the common out counts shows how well the rule of two and the rule of four hold up.

The one-card column is the out count divided by 46, the unseen cards after the turn. The two-card column is one minus the chance that neither the turn nor the river helps, which is 1 minus (38/47 x 37/46) for the 9 out row, giving 34.97 percent. The page prints 34.9 percent, which comes from rounding the 0.651 in that intermediate product to three decimal places before subtracting it. The rule of four overstates the two-card figure slightly and the rule of two understates the one-card figure slightly, and both stay within one and a half points across this range.

The flush draw against the page's pot odds

The page sets out a £100 pot and a £50 bet, so a call of £50 stands to win £150, which the page gives as pot odds of 3 to 1. Expressed as a required win rate, 3 to 1 means the call needs to win 50 / (150 + 50) = 25.00 percent of the time to break even, which is the 1 in 4 the page quotes.

The flush draw sits on both sides of that threshold depending on how many cards are left.

With two cards to come the draw has 34.97 percent against a 25.00 percent requirement, so the call returns £19.94 on average. With one card to come the equity of 19.57 percent falls short of the same requirement and the call loses £10.87 on average. The comparison turns on a fact the out count alone does not carry: the same 9 outs are worth more than 15 points more equity when two cards are still to come than when one is.

Worked the other way, the break-even pot odds are 1.86 to 1 with two cards to come and 4.11 to 1 with one. A pot offering 3 to 1 is therefore enough with two cards to come and not enough with one, which is the arithmetic the page's pot odds section describes.

What the odds do not cover

• The out count gives the chance of improving a hand, not the chance of winning the pot. The card that makes a flush can also make a full house for an opponent.
• The five-card counts assume a full 52-card deck with no wild cards. Jokers, a stripped deck or a five of a kind rule all change the denominator.
• The two-card percentage treats the hand as if both remaining cards are seen for one price. In a real game the turn brings another betting round, which is where the page's note on implied odds applies.
• The counts say nothing about opponents. Texas Hold'em equity depends on the range of hands an opponent holds, not only on your own outs.
• Ties are not modelled. A straight on the board can split the pot, and a split changes the value of a call without changing the out count.
• Five-card draw is the model throughout. In variants where players hold more or fewer cards, the deck and the denominator both change.

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