Probability of Poker Hands | Free Discrete Mathematics Lesson (4 College Credits) | UPI Study

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Probability of Poker Hands

Discrete Mathematics · Chapter 7: Probability in Games, Gambling, and Strategic DecisionSection 7.3 · lesson 42 of 8711 min · free with a free account

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In this lesson

Counting, combinations, and the mathematics behind every hand at the table.

  • Define the sample space of 5-card hands
  • Count every hand category with combinations
  • Convert counts to probabilities and odds
  • Explain the hand ranking as a ranking by rarity

Transcript

Everything said in the video, section by section (1,554 words). Read along, or search it.

Probability in Games, Gambling, and Strategic Decision

Probability of Poker Hands

This section counts poker hands. We'll define the sample space of five-card hands, count every hand category from first principles, and turn those counts into probabilities and odds. And we'll see that the traditional ranking of poker hands is simply the list sorted by rarity.

The Opening Problem

A Full House on the River

A player at an online table shows three tens and two fours on the river, and wins a large pot. The site's hand review tool labels it a full house, and quietly notes how unusual it was. There's no guesswork behind that label. A full house is rare because, of the two million, five hundred ninety-eight thousand, nine hundred sixty equally likely five-card hands, only three thousand seven hundred forty-four are full houses. Every probability on a poker training platform, and every equity percentage in analysis software, is a direct application of the counting methods from chapter five. This section derives that number, and the number for every other standard hand.

7.3.1 The Sample Space

2,598,960 Equally Likely Hands

A five-card hand is an unordered selection of five distinct cards, from a standard deck of fifty-two. That's thirteen ranks, in each of four suits. The order the cards arrive doesn't change the hand, so the sample space is every five-card subset of the deck. That's fifty-two choose five, two million, five hundred ninety-eight thousand, nine hundred sixty hands. With a properly shuffled deck, every one is equally likely. So the probability of any hand type is its count, over that total. Standard poker recognizes ten mutually exclusive hand categories. No hand belongs to more than one, so they split the sample space into ten disjoint pieces, whose sizes must add up exactly. Precise definitions matter here. An imprecise definition is exactly what causes double-counting.

Table 7.3.1

The 10 Categories, Precisely

Here are the ten categories, stated precisely. A royal flush is ten, jack, queen, king, ace, all one suit. A straight flush is any other same-suit sequence. Four of a kind is four cards of one rank, plus one other card. A full house is three of one rank and two of another. A flush is five cards of one suit that don't form a sequence. A straight is a sequence that isn't all one suit. Three of a kind has three of one rank, and two more cards of two different ranks. Two pair is two and two, plus a card of a third rank. And one pair is a pair, plus three cards of three other ranks. High card is everything else. Notice the exclusions. A same-suit ten through ace is a royal flush, not a flush and not a straight, because both of those definitions explicitly exclude it.

Full House - Fully Worked

4 Stages, Multiplied

Now the counts. Every one rests on the multiplication rule, with combinations for unordered choices. Start with the full house, a four-stage task. Stage one, choose the rank for the triple. Thirteen ways. Stage two, choose which three of that rank's four suits appear. Four choose three, which is four. Stage three, choose the rank for the pair. One rank is taken, leaving twelve. Stage four, choose two of that rank's four suits. Four choose two, which is six. Multiplying, thirteen times four times twelve times six is three thousand seven hundred forty-four. And thirteen, then twelve, isn't double-counting. Three kings and two sevens is a different hand from three sevens and two kings. The triple and the pair are distinguishable roles.

Flush and Straight

Count Everything, Then Remove the Overlap

For a flush, the cleanest route counts every same-suit hand first, then removes the overlap. Choose the suit, four ways, then five of its thirteen ranks, one thousand two hundred eighty-seven ways. That's five thousand one hundred forty-eight same-suit hands. But that still includes every straight flush, which has its own category. There are forty of those, so there are five thousand one hundred eight flushes. Straights work the same way. Counting the ace as high or low, a five-card sequence can start at ten different ranks, from ace-two-three-four-five up to ten through ace. Each of the five cards can be any of four suits, four to the fifth, one thousand twenty-four patterns. That's ten thousand two hundred forty sequences. Subtract the same forty straight flushes, and there are ten thousand two hundred straights.

3 of a Kind and 2 Pair

When a Stage Is Itself a Combination

Three of a kind. Choose the rank, thirteen ways, and three of its suits, four ways. Then choose two other ranks for the unmatched cards, from the remaining twelve. That's twelve choose two, sixty-six. And a suit for each, four times four, sixteen. That stage has to be a combination, not twelve then eleven. The two kicker ranks are unordered. A nine and a five is the same hand whichever is chosen first. Multiplying gives fifty-four thousand nine hundred twelve. Two pair uses the same idea. Choose the two pair ranks together, thirteen choose two, seventy-eight. Two suits for each, six times six, thirty-six. And the fifth card must avoid both pair ranks, leaving eleven ranks and four suits, forty-four. That's one hundred twenty-three thousand five hundred fifty-two.

The Rest

1 Pair, and the 3 Rarest Hands

One pair. Choose the rank, thirteen ways, and two of its suits, six ways. Then three other distinct ranks, twelve choose three, two hundred twenty. And a suit for each of those three, four cubed, sixty-four. That's one million, ninety-eight thousand, two hundred forty. The three rarest categories are quick. Four of a kind picks the rank, uses all four suits, and adds any of the forty-eight other cards. Six hundred twenty-four. Same-suit sequences pick a suit and one of ten starting ranks. Forty in all. Exactly four of them, one per suit, are ten through ace, the royal flushes. That leaves thirty-six straight flushes.

High Card - by the Complement Rule

Whatever Is Left

High card hands, with no pair, no sequence, and not all one suit, are awkward to count directly. Three conditions have to hold at once. The complement rule sidesteps this. The ten categories are mutually exclusive and exhaustive, so high card is whatever remains after the other nine. The nine named categories sum to one million, two hundred ninety-six thousand, four hundred twenty. Subtracting from the full sample space leaves one million, three hundred two thousand, five hundred forty high-card hands. As a check, all ten counts sum to exactly the sample space. No hand is left uncounted, and none is counted twice.

7.3.3 Counts to Probabilities

The Full House, 3 Ways

Turning counts into probabilities is now just division. Take the full house, three thousand seven hundred forty-four, over the total. Both numbers share a factor of six hundred twenty-four, so the fraction reduces to six over four thousand one hundred sixty-five. As a percentage, that's about zero point one four four percent. A number that small is easy to misjudge, so it's standard to also report the reciprocal, as one in N odds. The total divided by the count is about six hundred ninety-four. So a full house comes up about once in every six hundred ninety-four hands. The fraction, the percentage, and the odds all describe the identical probability.

Table 7.3.2

All 10 Hands, Rarest First

Repeating that for all ten categories gives this table, sorted from rarest to most common. A royal flush is about one in six hundred fifty thousand. A straight flush, about one in seventy-two thousand. Four of a kind, one in four thousand one hundred sixty-five. A full house, one in six hundred ninety-four. A flush, one in five hundred nine. A straight, one in two hundred fifty-five. Three of a kind, about one in forty-seven. Two pair, one in twenty-one. And at the bottom, one pair is about forty-two percent of hands, and high card about fifty percent.

Figure 7.3.2

5 Orders of Magnitude

The probabilities span more than five orders of magnitude, so this chart uses a logarithmic scale. Each bar's length is proportional to the logarithm of the count. A royal flush is roughly three hundred twenty-five thousand times rarer than a high-card hand. On an ordinary linear axis, the nine rarer categories would be indistinguishable from zero. Reading down, every step to a more common hand is a step down the ranking.

7.3.4 Why the Order Isn't Arbitrary

Rarer Ranks Higher

So why does a flush outrank a straight? There are five thousand one hundred eight flushes, and ten thousand two hundred straights, out of the same denominator. Dividing both by the same positive number keeps the inequality. So a flush is less likely than a straight. And a rarer hand ranks higher. So flush over straight isn't a separate design decision. It's forced by which count is smaller. Sorting all ten categories from rarest to most common produces the traditional ranking exactly. That's also why poker software uses these fixed probabilities as a starting point, a prior, before any cards are exposed. It then updates the estimate as betting action and community cards add information. That's conditional probability, which comes later.

Takeaways

Count, Divide, Sort

To summarize. The sample space has exactly fifty-two choose five equally likely hands, because a hand is an unordered, non-repeating selection. Every hand count follows from the multiplication rule across stages, with a combination whenever a stage is itself an unordered choice. Flush and straight each subtract the straight flushes, and the complement rule turns high card into a single subtraction. And the traditional ranking is exactly the categories sorted by rarity. Here's something to think about. Suppose the categories had been defined the easy way, as overlapping at least categories. At least a pair, at least two pair, at least a flush, and so on. What would happen to the sum of the ten counts? And why would the complement argument used for high card no longer be valid?

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FAQ

Questions students ask us

About Probability of Poker Hands

How many five-card poker hands are there, and why does order not matter?
There are \binom{52}{5} = 2{,}598{,}960 possible five-card poker hands. A hand is an unordered selection of five distinct cards, so the order of dealing doesn't change it. With a properly shuffled deck every hand is equally likely, so the probability of any hand type is its count divided by 2,598,960.
How do you count the number of full house hands and find the probability?
There are 3,744 full houses, found by multiplying four stages: 13\cdot\binom{4}{3}\cdot 12\cdot\binom{4}{2} = 13\cdot4\cdot12\cdot6. Choose the triple's rank, its three suits, the pair's rank from the remaining 12, then its two suits. Dividing by 2,598,960 gives 6/4165, about 0.144%, or roughly 1 in 694 hands.
Why do you use a combination for the two extra cards in three of a kind instead of 12 times 11?
Use \binom{12}{2} = 66 because the two unmatched ranks are unordered. A nine and a five is the same hand whichever you pick first, so 12\cdot 11 would count every hand twice. Three of a kind then has 13\cdot4\cdot66\cdot16 = 54{,}912 hands, where 16 counts the suit choices for the two extra cards.
Why does a flush rank higher than a straight in poker?
A flush ranks higher because it is rarer. There are 4\binom{13}{5}-40 = 5108 flushes and 10\cdot 4^5-40 = 10{,}200 straights, each subtracting the 40 straight flushes. Both counts share the same denominator, so a flush is less likely. Sorting all ten categories from rarest to most common reproduces the traditional ranking.

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