Poker Book, Poker Odds

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Appendix D
Poker Odds

Appendix D compiles the following card odds:

Rank of Hands with Odds
Draw Odds
Pat-Hand Odds
Lowball Odds
Hold 'em and Stud Odds
Seven-Stud Odds
Seven-Stud Catch Odds
Two-Pair Odds
Wild-Card Odds
Comparison of Odds.

Card odds can be calculated and expressed as shown below:

Example of Three-of-a-Kind Odds

Odds For	Deals Per Pat Hand	(52-card deck) Before the Draw	Lower Value Hands per Pat Hand	Odds Against
1 in 47	47	(Starting with 5 cards	46	46 to 1

Odds For	Draws per Catch	After the Draw	Misses per Catch	Odds Against
1 in 8.7	8.7	(Draw 3 cards to a pair)	7.7	7.7 to 1

Note: All values are rounded at two figures.

To calculate, for example, the number of three-of-a-kind hands possible on the deal, simply divide the deals per pat hand (47) into the total number of hands possible with a 52-card deck (2,598,960). That calculation gives a rounded-off answer of 55,300 possible hands of three of a kind on the deal. The precise answer (as shown in odds table #1 is 54,913 possible hands, which is calculated by using exact figures and not rounding off numbers.

But in calculating the card odds for drawing various poker hands (such as tabulated in odds table #2), a special problem arises that makes draw odds reported in all other poker books either inaccurate or imprecise. Furthermore, no practical way exists to give precise draw odds for certain hands. As a result, all the odds in this Appendix were defined and then calculated on IBM computers at the University of Delaware. Those calculations provided the only accurately defined and consistently calculated odds in the literature. While certain draw odds are not precise for every situation, all odds provided in this Appendix can be used with confidence since the additional knowledge of the slightly different, precise draw odds would probably never make a meaningful difference for any poker decision.

For those interested in a more detailed explanation of the draw-odds calculations, see the footnote for Poker-Odds Table #2.

1. RANK OF HANDS WITH ODDS

(highest to lowest)

Rank	Hand	Example	Number of Hands Possible
*	Five Aces (with Bug )	AAAAB	1⁺
*	Five of a Kind	8888W (joker wild)	13⁺
*	Five of a Kind (with Wild Card)	(deuces wild)	672
*	Skeet Flush	2S 4S 5S 8S 3S	24
1	Royal Straight Flush	10H JH QH KH AH	4
1	Straight Flush	4C 5C 6C 7C 8C	40
2	Four Aces	XAAAA	48
2	Four of a Kind	X7777	624
*	Big Bobtail	X 8D 9D JD QD	144
*	Blaze Full	QQKKK	144
3	Full House	66JJJ	3,744
4	Flush	DDDDD	5,108 (n.s.)
*	Big Tiger (Big Cat)	8 - - - K	4,096 (i.f.)
*	Little Tiger (Little Cat)	3 - - - 8	4,096 (i.f.)
*	Big Dog	9 - - - A	4.096 (i.f.)
*	Little Dog	2- - -7	4,096 (i.f.)
5	Straight	78910J	10,200 (n.f.)
*	Round the Corner Straight	32AKQ	3,060 (n.f.)
*	Skip Straight (Dutch Straight)	579JK	8,120 (n-f.)
*	Kilter	A - - - 9	35,840 (i.f.)
*	Five and Dime	5 - - -10	4,096
*	Skeet (Pelter, Bracket)	2 - 5 - 9	6,144 (i.f.)
6	Three of a Kind	XX10 10 10	54,912
*	Little Bobtail	XX6C 7C 8C	3,120
*	Flash	HDSCB	685,464 +
*	Blaze	PPPPP	792
7	Two Pair	X3399	123,552
*	Four Flush with a Pair	DDD 5D 5	34,320
*	Four Flush	XHHH	111,540
8	Pair	XXX88	1,098,240
9	No Pair (+)	XXXXX	1,302,540
9	Ace High (+)	- - - - A	502,860
9	King High (+)	- - - - K	335,580
9	Queen High (+)	- - - - Q	213,180
9	Jack High (+)	- - - - J	177,500
9	Ten High (+)	- - - - 10	70,360
10	Nine Low (+ +)	- - - - 9	71,860
10	Eight Low (+ +)	- - - - 8	35.840
10	Seven Low (+ +)	- - - - 7	15,360
10	Six Low (+ +)	- - - - 6	5,120
10	Five Low (+ +)	A2345	1,024

Total hands possible with a 52-card deck 2,598,960
⁺ Total hands possible with a 53-card deck (with a joker) 2,869,685

Code:
* = Not a normal hand (freak hand)
B = Bug card (joker)
W = Wild card
P = Any picture card
H = Heart
D = Diamond
S = Spade
C = Club
A = Ace
K = King
Q = Queen
J = Jack
X = Any nonpaired side card
- = A specific nonpaired side card
(+) = No straights or flushes, ace is high
(+ +) = Including straights and flushes, ace is low
i.f. = Including flushes,
n.f. = no flushes,
n.s. = no straights

2. DRAW ODDS

Original Hand	Cards Drawn	Final Hand	Approximate^* Draws per Catch
Ace	4	Two pair or better	14
Ace-King, same suit	3	Two pair or better	14
Pair	3	Any improvement	4
---	3	Two pair	6
---	3	Trips	9
---	3	Full	100
---	3	Four	380
Two-card flush	3	Flush	100
Pair + kicker	2	Any improvement	4
---	2	Two pair	6
---	2	Trips	13
---	2	Full	125
---	2	Four	1100
Pair + ace	2	Aces up	9
---	2	Two pair (lower)	18
Trips	2	Any improvement	10
---	2	Full	16
---	2	Four	24
Three-card straight flush, double open	2	Straight or better	12
---	2	Straight flush	1100
Three-card straight flush, KQJ or 432	2	Straight or better	14
Three-card straight flush, AKQ or 32A	2	Straight or better	21
Three-card straight, double open	2	Straight	24
Three-card flush	2	Flush	25
Two pair	1	Full	12
Trips + kicker	1	Any improvement	12
---	1	Full	l6
---	1	Four	48
Four-card straight, open both ends	1	Straight	6
Four-card straight, inside or one end	1	Straight	12
Four-card flush	1	Flush	5
Four-card straight flush, open both ends	1	Straight or better	3
---	1	Straight flush	24
Four-card straight flush, inside or one end	1	Straight or better	4
---	1	Straight flush	48

* Approximate values rather than precise values must be reported for the following reason: Consider an extreme example--the odds on a four-card draw to an ace. Does one assume a blind draw into a forty-seven-card deck that would give a precise value of 12.8 draws per catch of two pair or better? Or does one assume a draw into a fifty-one-card deck (a deck with one ace missing) that would give a precise value of 15.6 draws per catch of two pair or better? Now a 20 percent difference exists between those two precise values with no basis for selecting one assumption over the other (forty-seven-card deck versus fifty-one-card deck). Furthermore, neither assumption represents the actual situation: The draw is not blind from a forty-seven-card deck, and the draw is not from a fifty-one-card deck. An accurate and precise value is obtained only by defining each of the four discarded cards and then drawing from a forty-seven-card deck. But that would not be practical because a complete table of draw odds to the ace alone would consist of hundreds of thousands of values. All those values do, however, lie somewhere between the values for a blind draw into the forty-seven-card deck and a draw into the fifty-one-card deck. So where necessary, draw odds are calculated at the midway value between the two extreme precise values and then rounded off to a whole number. That is the most practical way to report such draw odds in a consistent and accurately defined manner.

3. PAT-HAND ODDS

A. Various Hands

Hand	Hands Possible	Pat Hands per 200,000 Deals	Deals per Pat Hand	Deals per Pat Hand or Better
Royal straight flush	4	.15	649,740	649,740
Straight flush	36	1.4	72,193	64,974
Four of a kind	624	22	4,165	3,914
Full house	3,744	144	694	590
Flush	5,108	196	509	273
Straight	10,200	392	255	132
Three of a kind	54,912	2,113	47	35
Two pair	123,552	4,754	21	13
One pair	1,098,240	42,257	2.4	2
No pair	1,302,540	50,118	2	1
Total	2,598,960	100,000	---	---

B. High Pairs

Hand	Hands Possible	Pat Hands per 200,000 Deals	Deals per Pat Hand	Deals per Pat Hand or Better
Aces	84,480	3,250	31	9
Kings	84,480	3,250	31	7
Queens	84,480	3,250	31	6
Jacks	84,480	3,250	31	5

C. Draw Hands to Straights and Flushes

(Compiled for the Advanced Concepts of Poker
by Michael J. Caro, a leading authority
on draw poker and poker mathematics.)

Hand	Hands Possible	Pat Hands per 200,000 Deals	Deals per Pat Hand	Deals per Pat Hand or Better
Four-card straight,^* any	325,008	12,505	8	---
Four-card straight,^* inside	251,136	9,663	10	---
Four-card straight,^* outside	73,872	2,842	35	---
Four-card flush^*	105,744	4,068	25	---
Four-card straight flush^*	5,796	223	448	---
Three-card straight flush^*	8,064	310	322	---

^* Excludes pat hands and higher-value draws.

4. LOWBALL ODDS

A. Pat Card Odds on the Deal

(52-card deck--no joker^*)

Pairless Hands Possible

Highest Card in five cards	Including Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is High
Ace	0	0	502,860
King	508,880	502,860	335,580
Queen	337,920	335,580	213,180
Jack	215,040	213,180	127,500
Ten	129,024	127,500	70,360
Nine	71,680	70,360	34,680
Eight	35,840	34,680	14,280
Seven	15,360	14,280	4,080
Six	5,120	4,080	0
Five	1,024	0	0

B. Draw Odds

(52-card deck--no joker^*)

One-Card Draws per Catch

Highest Card in four cards	Highest Card in five cards	Including Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is High
Ten	Ten	2	2.03	2.45
Nine	Nine	2.4	2.45	3.10
Eight	Eight	3	3.10	4.30
Seven	Seven	4	4.30	7.53
Six	Six	6	7.53	---
Five	Five	12	---	---

Two-Card Draws per Catch

Highest Card in three cards	Highest Card in five cards	Including Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is High
Eight	Eight	7.35	7.59	13.44
Seven	Seven	12.50	13.44	30.75
Six	Six	24.50	30.75	---
Five	Five	73.50	---	---

Three-Card Draws per Catch

Highest Card in two cards	Highest Card in five cards	Including Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is High
Seven	Seven	30.63	31.91	95.15
Six	Six	76.56	95.15	---
Five	Five	306.25	---	---

Four-Card Draws per Catch

Highest Card in one cards	Highest Card in five cards	Including Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is Low	No Straights and Flushes, Ace is High
Seven	Seven	65.08	70.02	244.96
Six	Six	195.24	244.96	---
Five	Five	976.17	---	---

^* For a fifty-three-card deck with a joker, the number of pat hands possible increases by a few percent to several hundred percent, depending on the hand.

5. HOLD 'EM AND STUD ODDS

First Two Cards ---------- (stud and hold 'em)	Deals per Catch
2 aces	221
2 kings, etc.	221
Any pair	17
Any hand with a pair or an ace	5
Ace-king suited	332
Ace-king not suited	111
Any two cards suited	4

First Three Cards (High) ---------- (7 stud & pineapple hold 'em)	Deals per Catch
3 aces	5525
3 kings, etc.	5525
Any trips	425
Three straight flush	86
Three flush	25
2 aces	77
Any pair	6

First Three Cards (Low) ---------- (Razz)	Deals per Catch
A-2-3 (lowest)	345
4 and lower	86
5 and lower	34
6 and lower	17
7 and lower	10
8 and lower	6
9 and lower	4

6. SEVEN-STUD ODDS

Hand	Hands Possible	Approximate Hands per 100,000 Deals
Straight flush	37,444	28
Four of a kind	224,848	168
Full house	3,473,184	2,590
Flush	4,051,784	3,030
Straight	8,466,876	6,330
Three of a kind	6,374,520	4,760
Two pair	30,834,000	23,050
One pair	56,851,296	42,500
No pair	23,470,608	17,500
Total Hands	133,784,560

7. SEVEN-STUD CATCH ODDS

Start With	Misses per Catch of a Straight (outside)	Misses per Catch of a Flush	Misses per Catch of a Full House or Fours
FFX	66	31	13
FFXX	106	275	19
FFXXX	--	---	38
FFGG	53	137	4
FFGGX	---	---	7
FFGGXX	---	---	11
FFF	4	4.5	1.5 (11 for fours)
FFFX	8	9	1.7
FFFXX	22	23	2
FFFXXX	---	---	4
FFFF	1.5	1	---
FFFFX	2	2	---
FFFFXX	5	4	---

F or G = a flush, straight (outside) or a paired card.
X = a nonhelping card.

8. TWO-PAIR ODDS

Hand	Hands Possible	Hands Higher	Hands Lower
Aces up	19,008	0	104,544
Kings up	17,424	19,008	87,120
Queens up	15,840	36,432	71,280
Jacks up	14,256	52,272	57,024

----------------------------50%----------------------------

Tens up	12,672	86,528	44,352
Nines up	11,088	79,200	33,264
Eights up	9,504	90,288	23,760
Sevens up	7,920	99,792	15,840
Sixes up	6,336	107,712	9,504
Fives up	4,752	114,048	4,752
Fours up	3,168	118,800	1,582
Threes up	1,584	121,968	0
Total	123,552	---	---

9. WILD-CARD ODDS

Various Hands

Deals to Get on First Five Cards

Hand	No Wild Cards	Joker Wild	Deuces Wild	Deuces Wild, Hands Possible
Five of a kind	--	220,745	3,868^*	672
Royal straight flush	649,740	119,570	5,370	484
Straight flush	72,193	14,666	575	4,072
Four of a kind	4,165	920	81^*	30,816
Full house	694	438	205	12,672
Flush	509	362	159	13,204
Straight	255	221	38	66,236
Trips	47	21^*	8^*	355,056
Two pair	21	23	27	95,040
One pair	2.4	2.4	2.4	1,222,048
No pair	2	2.2	3.4	798,660
Total	---	---	---	2,598,960

^* With deuces wild, five of a kind is easier to get than a straight flush, four of a kind is much easier to get than a flush or a full house, and three of a kind is easier to get than two pair.