Probability of Low-Hand Poker Variants | Free Discrete Mathematics Lesson (4 College Credits) | UPI Study

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Probability of Low-Hand Poker Variants

Discrete Mathematics · Chapter 7: Probability in Games, Gambling, and Strategic DecisionSection 7.4 · lesson 43 of 879 min · free with a free account

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In this lesson

Same cards, different rules - and 2 different ways to invert a ranking.

  • Distinguish Ace-to-Five from Deuce-to-Seven lowball
  • Compare 2 low hands with a precise procedure
  • Compute low-hand probabilities from C(52, 5)
  • See why ranking code must encode its convention

Transcript

Everything said in the video, section by section (1,316 words). Read along, or search it.

Probability in Games, Gambling, and Strategic Decision

Probability of Low-Hand Poker Variants

This section covers lowball poker, where the lowest hand wins. There are two main ranking systems, and each inverts the standard ranking from section seven point three in a genuinely different way. We'll pin both down precisely, compare hands under one of them, and compute probabilities specific to each.

The Opening Problem

7-5-4-3-2: Worthless or Unbeatable?

At a mixed-game table, a player raised on standard hold 'em is dealt seven, five, four, three, two. Unsuited, with no pair. In a normal poker hand, that's nearly worthless. Seven-high, nothing connects. A fold in almost any situation. In a Razz hand at the same table, those same five ranks would be one card away from the best possible hand in the game. And in a Deuce-to-Seven hand, that exact holding is the best possible hand. Unbeatable. The cards never changed. Only the rule for what counts as winning did.

Definition 7.4.1

Ace-to-Five ("California") Lowball

Lowball isn't one ranking system. It's a family of games where the lowest-ranked hand wins. Two conventions dominate real play. The first is Ace-to-Five, sometimes called California lowball. It's used in Razz, a seven-card stud game, and in Ace-to-Five draw. Here, straights and flushes aren't recognized at all. They neither help nor hurt. And the ace always plays as the lowest card. Any hand with a pair, or anything better, is worse than every hand with five distinct ranks, whatever the values. Among pair-free hands, compare the highest cards first. If they tie, compare the next highest, and so on. The best possible hand is ace, two, three, four, five, called the wheel.

Definition 7.4.2

Deuce-to-Seven ("Kansas City") Lowball

The second is Deuce-to-Seven, sometimes called Kansas City lowball. It's used in two-seven single draw, and two-seven triple draw. Here, straights and flushes are recognized exactly as in standard poker. But they count against a hand. The whole category hierarchy from section seven point three is kept. Only the winning direction is reversed. And the ace always plays high, never low. So ace through five isn't a straight here. The best possible hand is seven, five, four, three, two, unsuited. It's the lowest set of five distinct ranks that avoids a straight, since two through six would be one.

Table 7.4.1

2 Different Inversions

This table sets the two conventions against standard poker. Standard poker is won by the highest hand. Both lowball conventions are won by the lowest. Straights and flushes help a standard hand, aren't recognized at all in Ace-to-Five, and hurt a hand in Deuce-to-Seven. The ace is high in standard poker, except in a five-high straight. It's always low in Ace-to-Five, and always high in Deuce-to-Seven. So Ace-to-Five discards categories and redefines the ace. Deuce-to-Seven keeps everything, and flips only the direction of winning.

Figure 7.4.1

Flipped Versus Collapsed

Here's that contrast as a picture. Standard poker's ten-category hierarchy, on the left, reappears unchanged in Deuce-to-Seven, in the middle. Only the winning end moves, from the top to the bottom. Ace-to-Five, on the right, is structurally different. It collapses the hierarchy to a single split. Pair-free hands, which win, against pair-or-better hands, which lose.

Worked Example 7.4.1

Comparing 2 Ace-to-Five Hands

Comparing two Ace-to-Five hands is a two-stage procedure. Compare the eight, six, four, three, two, with the eight, six, four, three, ace. First, check for pairs. Both hands have five distinct ranks. The ace is its own rank. So both are pair-free, and must be compared card by card. Second, sort each hand from highest to lowest, with the ace valued at one. Then compare position by position. Eight ties eight, six ties six, four ties four, three ties three. In the last position, two against one. Lower wins, so the hand with the ace wins. The comparison is positional, not a sum. The totals are irrelevant, because Ace-to-Five never adds card values. It compares them, in order. And the category check dominates whenever it applies. Two, two, three, four, five has a pair, so it loses to six, five, four, three, two, even though the six is higher. Category beats raw value.

Code Example 7.4.1

The Procedure, Codified

That procedure codes up directly. First, check whether each hand has a pair or better, meaning fewer than five distinct ranks. If exactly one does, the pair-free hand wins. Otherwise, sort both hands in descending order, with the ace as one, and compare position by position. The lower value wins the first position that differs. Run on the worked example, it returns B, the hand with the ace, confirming the derivation.

Worked Example 7.4.2

P(Dealt the Wheel)

The sample space is still fifty-two choose five. It depends only on the deck, not on the ranking rules. What changes is which subsets are worth counting. What's the probability of being dealt the wheel, in any suits? It needs exactly one ace, two, three, four, and five. Each of those five ranks can be any of four suits. That's four to the fifth, one thousand twenty-four hands. Divided by the total, it's sixty-four over one hundred sixty-two thousand, four hundred thirty-five. About one hand in two thousand five hundred thirty-eight. And every other specific pattern of five distinct ranks has exactly the same probability. A smooth six, or a rough seven. The derivation never used anything special about the ranks one through five. What differs between patterns is how each ranks once dealt, not how likely it is.

Worked Example 7.4.3

How Often Is a Hand Pair-Free?

A broader question. How often is a hand pair-free at all, so it qualifies for that comparison? Choose five distinct ranks, thirteen choose five, one thousand two hundred eighty-seven ways. Then a suit for each, one thousand twenty-four ways. That's one million, three hundred seventeen thousand, eight hundred eighty-eight hands. A probability of about fifty point seven percent. That's a little larger than the high-card probability from section seven point three. Here, straight-shaped and flush-shaped hands with five distinct ranks count too, since Ace-to-Five doesn't recognize those categories. By the complement rule, about forty-nine point three percent of hands contain a pair or better. Roughly half of all hands are out of contention before a single card is compared.

Worked Example 7.4.4

P(Dealt 7-5-4-3-2) in Deuce-to-Seven

In Deuce-to-Seven, straights and flushes do exist, so a probability question has to account for both. What's the probability of being dealt seven, five, four, three, two, in any suits that aren't a flush? First, confirm it's not a straight. The ranks skip six, so it isn't. Second, count suit assignments. As before, one thousand twenty-four. Third, remove the flushes. Exactly four assignments are all one suit, which is worse in Deuce-to-Seven. That leaves one thousand twenty. So the probability is one in two thousand five hundred forty-eight. Almost identical to the wheel, though they're the best hands under two different rule sets.

Encoding the Convention

One Ranking Function Won't Do

This is exactly the kind of logic that hand-ranking software must encode correctly, rather than infer. A ranking module written for Ace-to-Five can't simply be reused for Deuce-to-Seven. Its treatment of the ace would need inverting. Its straight and flush exclusions would need reinstating. And its best-hand target would change, from ace through five, to seven, five, four, three, two. So software supporting mixed-game formats must hard-code which convention applies to each game, and test each one independently. A single mis-encoded ace value is exactly the kind of silent, hard-to-notice bug that merging the two conventions would introduce.

Takeaways

Same Deck, Different Questions

To summarize. Lowball isn't a single system. Ace-to-Five ignores straights and flushes, with the ace always low. Deuce-to-Seven keeps the full hierarchy, reverses only which end wins, and keeps the ace high. Comparing Ace-to-Five hands is category first, then position by position from the highest card down. The sample space doesn't change between conventions. Only which subsets are meaningful does. And the probability of a specific rank pattern doesn't depend on how favorably it's ranked. Here's something to think about. A player with years of standard poker experience sits down in a Razz game, and is dealt six, four, three, two, ace. They fold at once, reasoning that it's ace-high, and not even a pair. Which assumption fails to carry over from standard poker? What hand were they actually holding under Ace-to-Five rules, and how strong is it compared with the wheel?

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FAQ

Questions students ask us

About Probability of Low-Hand Poker Variants

What is the difference between Ace-to-Five and Deuce-to-Seven lowball?
Ace-to-Five ignores straights and flushes and always plays the ace low, while Deuce-to-Seven keeps standard poker's full hand categories and plays the ace high. In Ace-to-Five the best hand is A-2-3-4-5, the wheel. In Deuce-to-Seven, straights and flushes count against a hand, so the best hand is 7-5-4-3-2 unsuited.
How do you compare two Ace-to-Five lowball hands?
Check for pairs first: any hand with a pair or better loses to every hand with five distinct ranks. If both hands are pair-free, sort each from high to low with the ace as 1, then compare position by position. The first difference decides, and the lower card wins. So 8,6,4,3,A beats 8,6,4,3,2. Sums are irrelevant.
What is the probability of being dealt the wheel, or 7-5-4-3-2 in Deuce-to-Seven?
Each of the five ranks can come in any of 4 suits, giving 4^5=1024 hands out of \binom{52}{5}=2{,}598{,}960. The wheel has probability 64/162435, about 1 in 2,538. For 7-5-4-3-2 in Deuce-to-Seven, remove the 4 flushes, leaving 1020 hands, or 1 in 2,548.
Why can't one hand-ranking function handle both Ace-to-Five and Deuce-to-Seven?
The two conventions differ in the ace's value, in whether straights and flushes count, and in which hand is best, so code for one gives wrong answers for the other. Reusing Ace-to-Five code for Deuce-to-Seven would require inverting the ace, reinstating straight and flush penalties, and changing the best-hand target. Each convention should be hard-coded and tested separately.

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