HEARTLESS POKER DOMINIC LANPHIER AND LAURA TAALMAN A. The probabilities, and hence the rankings, of the standard poker hands are well-known. We study what happens to the rankings in a game where a deck is used with a suit missing (heartless poker, for example), or with an extra suit, or extra face cards. In particular, does it ever happen that two or more hands will be equally likely? In this paper we examine this and other questions, and show how probability, some analysis, and even number theory can be applied. 1. S,F, FH? In standard 5-card poker, some hands are obviously better than others. Even the novice player will easily recognize that Four of a Kind is exciting, while One Pair is comparatively weak. Some players, however, may occasionally need to be reminded which of Full House, Flush, and Straight is the most valuable. Without computing probabilities it is not immediately obvious which of these three hands is the least likely, and thus ranked the highest. The rankings of these three hands in a regular poker deck with 13 ranks (2-10 and J, Q, K, A) and four suits (spades ♠, hearts , clubs ♣, and diamonds ) are shown in Figure 1. Straight Flush Full House 4 ♠ ♠ 4 ♠ ♠ ♠ ♠ 5 5 6 6 7 ♣ ♣ 7 ♣ ♣ ♣ ♣ ♣ ♣ ♣ 8 8 < 3 3 5 5 9 9 J J K K < Q ♣ ♣ Q ♣ ♣ Q Q Q ♠ ♠ Q ♠ ♠ 4 4 4 4 F1. A Straight is five cards with ranks in sequence. A Flush is five cards of the same suit. A Full House is three cards of one rank and two cards of another rank. Card images and layout from [2]. Of course it is not difficult to actually compute the necessary probabilities. To build a Full House you must choose a rank, and then three cards in that rank; then you must choose a second rank, 1
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HEARTLESS POKER
DOMINIC LANPHIER AND LAURA TAALMAN
A. The probabilities, and hence the rankings, of the standardpoker hands are well-known. We
study what happens to the rankings in a game where a deck is used with a suit missing (heartless poker,
for example), or with an extra suit, or extra face cards. In particular, does it ever happen that two or
more hands will be equally likely? In this paper we examine this and other questions, and show how
probability, some analysis, and even number theory can be applied.
1. S, F, F H?
In standard 5-card poker, some hands are obviously better than others. Even the novice player will
easily recognize that Four of a Kind is exciting, while One Pair is comparatively weak. Some players,
however, may occasionally need to be reminded which of Full House, Flush, and Straight is the most
valuable. Without computing probabilities it is not immediately obvious which of these three hands
is the least likely, and thus ranked the highest. The rankings of these three hands in a regular poker
deck with 13 ranks (2-10 and J, Q, K, A) and four suits (spades♠, heartsr, clubs♣, and diamonds
q) are shown in Figure 1.
Straight Flush Full House
4♠
♠4
♠
♠
♠
♠
5r
r5
r
r
r
r
r
6r
r6
r
r
r
r
r
r
7♣
♣7
♣♣
♣
♣♣
♣
♣
8q
q8
q
q
q
q
q
q
q
q <
3r
r3
r
r
r
5r
r5
r
r
r
r
r
9r
r9
r
r
r
r
r
r
r
r
r
Jr
rJ
r
r
Kr
rK
r
r
<
Q♣
♣Q
♣
♣
Qr
rQ
r
r
Q♠
♠Q
♠
♠
4r
r4
r
r
r
r
4q
q4
q
q
q
q
F 1. A Straightis five cards with ranks in sequence. AFlush is five cards of
the same suit. AFull Houseis three cards of one rank and two cards of another
rank. Card images and layout from [2].
Of course it is not difficult to actually compute the necessary probabilities. To build a Full House
you must choose a rank, and then three cards in that rank; thenyou must choose a second rank,1
2 DOMINIC LANPHIER AND LAURA TAALMAN
and then two cards in that rank. Since there are 13 possible ranks and 4 possible suits (hearts,
clubs, spades, diamonds) in a standard deck of cards, the number of ways to build a Full House is(
131
)(
43
)
·(
121
)(
42
)
= 3744. If we divide this number of possible Full House hands bythe total number(
525
)
= 2, 598, 960 of possible hands, we see that the probability of being dealt a Full House from a
standard deck is64165 ≈ 0.00144.
Similar calculations produce the well-known list of the frequencies and probabilities of poker
hands shown in Table 1. Note that each poker hand is represented exactly once in this table; for
example, the number of Flushes excludes those hands that areStraight Flushes, and the number of
Straight Flushes excludes Royal Flushes. The last column isthe ratio of the frequency of a given
poker hand divided by the frequency of the next higher rankedhand. For example, the 6 in the Full
House row asserts that a Full House is 6 times more likely thana Four of a Kind. Similarly, a player
is 17.33 times more likely to get a Four of a Kind than a Straight Flush.
Poker Hand Calculation Frequency Probability Ratio
Royal Flush(
41
)
4 0.00000154 -
Straight Flush(
101
)(
41
)
−(
41
)
36 0.00001385 9
Four of a Kind(
131
)(
44
)
·(
481
)
624 0.00024010 17.333
Full House(
131
)(
43
)
·(
121
)(
42
)
3,744 0.00144058 6
Flush(
135
)(
41
)
−(
101
)(
41
)
5,108 0.00196540 1.364
Straight(
101
)(
41
)5−
(
101
)(
41
)
10,200 0.00392465 1.997
Three of a Kind(
131
)(
43
)
·(
122
)(
41
)254,912 0.02112845 5.384
Two Pair(
132
)(
42
)2·(
111
)(
41
)
123,552 0.04753902 2.25
One Pair(
131
)(
42
)
·(
123
)(
41
)31,098,240 0.42256902 8.889
High Card all other hands 1,302,540 0.50117739 1.186
T 1. Frequency and probability of hands in standard 5-card poker.
The common confusion about the rankings of Full House, Flush, and Straight is understandable;
being dealt a Flush is just 1.997 times more likely than beingdealt a Straight, and being dealt a Full
House is just 1.364 times more likely than being dealt a Flush. To see this graphically, consider the
logarithmically scaled plot of hand frequencies shown in Figure 2. Full House, Straight, and Flush
are clearly clustered together compared to the other hands.Because of this clustering, we will focus
on these three hands in this paper.
HEARTLESS POKER 3
RF SF 4K FH F S 3K 2P 1P HCHand
10
100
1000
10000
100000
1000000
Frequency
F 2. Logarithmic plot of poker frequencies
2. M L: F G P H
Poker is typically played with a standard deck of 52 cards with 13 ranks and 4 suits. There are many
variations of poker by changing various rules such as how many cards are dealt, how many can be
traded, and whether any cards are wild. We could also vary thedeck itself, say by removing certain
cards at the outset.
For example, the game ofHeartless Pokeruses a standard deck with all the hearts removed. The
fact that there are only three suits and 39 cards in the deck changes the probabilities, and perhaps the
rankings, of the different types of hands. With only three suits we would expect itto be much easier
to get a Flush in Heartless Poker than with the usual deck. Moreover, with only three cards of each
rank, we might expect it to be more difficult than usual to get a Straight. In fact, in a moment we will
see that in Heartless Poker the rankings of Flush and Straight are reversed from their usual rankings;
a Heartless Poker Straight is more valuable than a HeartlessPoker Flush, as shown in Figure 3.
Heartless Flush Heartless Straight Heartless Full House
3q
q3
q
q
q
5q
q5
q
q
q
q
q
9q
q9
q
q
q
q
q
q
q
q
q
Jq
qJ
q
q
Kq
qK
q
q
<
4♠
♠4
♠
♠
♠
♠
5♣
♣5
♣
♣
♣
♣
♣
6♣
♣6
♣♣
♣
♣♣
♣
7♣
♣7
♣♣
♣
♣♣
♣
♣
8q
q8
q
q
q
q
q
q
q
q <
Q♣
♣Q
♣
♣
Qq
qQ
q
q
Q♠
♠Q
♠
♠
4♣
♣4
♣
♣
♣
♣
4q
q4
q
q
q
q
F 3. In Heartless Poker, the relative rankings of the Flush andStraight hands
are reversed.
4 DOMINIC LANPHIER AND LAURA TAALMAN
We could also modify the game of poker by increasing the size of the deck. For example, the
commercially availableFat Packdeck of cards has the standard 13 possible ranks, but each rank
appears ineightsuits (the usual spades, hearts, clubs, and diamonds along with the new suits tridents
È, roses✿, hatchetsO, and dovese). With so many cards in each rank we might expect it to be much
easier to get a Straight, and that is in fact the case. Interestingly, as we will see in a moment, it is
comparatively easy to get a Full House in Fat Pack Poker, but difficult to get a Flush. In other words,
in Fat Pack Poker the rankings of Full House and Flush are reversed from their usual positions, as
illustrated in Figure 4.
Fat Pack Straight Fat Pack Full House Fat Pack Flush
4È
È4
È
È
È
È
5r
r5
r
r
r
r
r
6e
e
6
e
e
e
e
e
e
7♣
♣7
♣♣
♣
♣♣
♣
♣
8✿
✿8
✿
✿
✿
✿
✿
✿
✿
✿ <
Q♣
♣Q
♣
♣
QO
OQ
O
O
Qe
e
Q
e
e
4r
r4
r
r
r
r
4È
È4
È
È
È
È
<
3È
È3
È
È
È
5È
È5
È
È
È
È
È
9È
È9
È
È
È
È
È
È
È
È
È
JÈ
ÈJ
È
È
KÈ
ÈK
È
ÈF 4. When playing with theFat Packdeck, the relative rankings of Flush and
Full House hands are reversed.
Basic counting arguments give the poker hand frequencies inTable 2 for decks withs suits and
r ranks. The entries for Straights and Straight Flushes hold only for r ≥ 6, since forr = 5 the only
possible Straight isA 2 3 4 5 (regardless of whether Ace is considered high or low) and for r < 5 it
is impossible to construct a Straight hand. Note that the sumof the numbers of hands is(
rs5
)
−(
r1
)(
s5
)
,
due to the possibility of obtaining the non-valid poker hand“5 of a kind” whens≥ 5.
The general frequency formulas in Table 2 allow us to prove a small preliminary result:
Theorem 1. Every possible permutation of Straight, Flush, and Full house rankings occurs for some
deck with s suits and r ranks.
Theorem 1 follows immediately by simply applying the frequency formulas in Table 2 to the six
(r, s) examples in Table 3. The first row is the usual 4-suit, 13-rank poker deck, the second row is the
3-suit, 13-rank Heartless Poker deck, and the third row is the 8-suit, 13-rank Fat Pack deck.
HEARTLESS POKER 5
Poker Hand Number of possible handsLeading term
Royal Flush(
s1
)
s
Straight Flush(
r−31
)(
s1
)
−(
s1
)
rs
Four of a Kind(
r1
)(
s4
)(
rs−s1
)
r2s5
Full House(
r1
)(
s3
)(
r−11
)(
s2
)
r2s5
Flush(
r5
)(
s1
)
−(
r−31
)(
s1
)
r5s
Straight(
r−31
)(
s1
)5−
(
r−31
)(
s1
)
rs5
Three of a Kind(
r1
)(
s3
)(
r−12
)(
s1
)2r3s5
Two Pair(
r2
)(
s2
)2(r−21
)(
s1
)
r3s3
One Pair(
r1
)(
s2
)(
r−13
)(
s1
)3r4s4
High Card[
(
r5
)
− (r − 3)][
(
s1
)5− s
]
r5s5
T 2. Frequencies of the possible poker hands for a deck withs suits andr
ranks per suit. The leading terms in the last column are relevant to the next section.
(r, s) Straight Flush Full House Ranking
(13, 4) 10,200 5,108 3,744 S < F < H
(13, 3) 2,400 3,831 468 F < S < H
(13, 8) 327,600 10,216 244,608 S < H < F
(25, 15) 16,705,920 796,620 28,665,000 H < S < F
(30, 7) 453,600 997,353 639,450 F < H < S
(33, 9) 1,771,200 2,135,754 3,193,344 H < F < S
T 3. Examples of frequencies and rankings of Straight (S), Flush (F), and
Full House (H) for decks withssuits andr cards per suit. Note that hands with the
lowest frequencies are ranked highest.
3. A S, F, F H E T?
We now come to the main question of this paper: are there any generalized poker decks for which
Straight, Flush, and Full House hands have the same frequency, either pairwise or all together? We
will handle the latter part of the question in this section and the pairwise question in the next.
6 DOMINIC LANPHIER AND LAURA TAALMAN
From the leading terms in the last column of Table 2 we see thatthe relative rankings of Full
House and Flush will switch infinitely often asr grows and then assgrows, and so on. However, this
is not the case for the relative rankings of Full House and Straight; although Full House is ranked
higher than Straight in standard poker, we will see that thisoccurs only in a finite number of cases.
To aid in our search for pairs (r, s) of rank and suit sizes for which there are ties between hands, we
investigate three curves and their intersections.
Definition 1. For r ≥ 6 and s≥ 1, define the following three curves: