Texas Holdem Poker Full House

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Introduction

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A Full House is any three cards of the same number or face value, plus any other two cards of the same number or face value. An example of a full house is, but is a full house as well. I'm creating a console texas hold'em poker. I'm already done making this game, everything works as supposed, expect for a full house for which I became undecided for a best way to write a code. This is how I present cards: 'D5', 'S2', 'SA'. The Full House is third on the list of poker hand rankings.It is made up of 3 same-ranked cards paired with 2 same-ranked cards. However, it's not as simple to figure out what a Full House means just from its name.

This page examines the probabilities of each final hand of an arbitrary player, referred to as player two, given the poker value of the hand of the other player, referred to as player one. Combinations shown are out of a possible combin(52,5)×combin(47,2)×combin(45,2) = 2,781,381,002,400. The primary reason for this page was to assist with bad beat probabilities in a two-player game, for example the Bad Beat Bonus in Ultimate Texas Hold 'Em.

For example, if you wish to know the probability of a particular player getting a full house and losing to a four of a kind, we can see from table 7 that there are 966,835,584 such combinations. The same table shows us that given that player one has a full house, the probability of losing to a four of a kind is 0.013390. To get the probability before any cards are dealt, divide 966,835,584 by the total possible combinations of 2,781,381,002,400, which yields 0.0002403.

Table 1 shows the number of combinations for each hand of a second player, given that the first player has less than a pair.

Table 1 — First Player has Less than Pair

Event	Pays	Probability
Less than pair	164,934,908,760	0.340569
Pair	228,994,769,160	0.472845
Two pair	43,652,558,880	0.090137
Three of a kind	7,303,757,580	0.015081
Straight	26,248,866,180	0.054201
Flush	13,060,678,788	0.026969
Full house	-	0.000000
Four of a kind	-	0.000000
Straight flush	85,751,460	0.000177
Royal flush	10,532,592	0.000022
Total	484,291,823,400	1.000000

Table 2 shows the number of combinations for each hand of a second player, given that the first player has a pair.

Table 2 — First Player has a Pair

Event	Pays	Probability
Less than pair	228,994,769,160	0.187874
Pair	574,484,133,960	0.471324
Two pair	270,127,833,552	0.221621
Three of a kind	47,736,401,832	0.039164
Straight	50,797,137,096	0.041676
Flush	30,076,271,352	0.024675
Full house	15,829,506,000	0.012987
Four of a kind	586,278,000	0.000481
Straight flush	214,250,184	0.000176
Royal flush	25,380,864	0.000021
Total	1,218,871,962,000	1.000000

Table 3 shows the number of combinations for each hand of a second player, given that the first player has a two pair.

Table 3 — First Player has a Two Pair

Event	Pays	Probability
Less than pair	43,652,558,880	0.066798
Pair	270,127,833,552	0.413355
Two pair	246,286,292,328	0.376872
Three of a kind	31,155,189,408	0.047674
Straight	18,549,991,152	0.028386
Flush	14,200,694,712	0.021730
Full house	28,751,944,680	0.043997
Four of a kind	653,378,400	0.001000
Straight flush	109,829,304	0.000168
Royal flush	12,673,584	0.000019
Total	653,500,386,000	1.000000

Table 4 shows the number of combinations for each hand of a second player, given that the first player has a three of a kind.

Table 4 — First Player has a Three of a Kind

Event	Pays	Probability
Less than pair	7,303,757,580	0.054369
Pair	47,736,401,832	0.355348
Two pair	31,155,189,408	0.231918
Three of a kind	27,586,332,384	0.205352
Straight	3,310,535,196	0.024643
Flush	2,606,403,900	0.019402
Full house	12,910,316,760	0.096104
Four of a kind	1,705,867,680	0.012698
Straight flush	19,970,844	0.000149
Royal flush	2,304,216	0.000017
Total	134,337,079,800	1.000000

Table 5 shows the number of combinations for each hand of a second player, given that the first player has a straight.

Table 5 — First Player has a Straight

Event	Pays	Probability
Less than pair	26,248,866,180	0.204299
Pair	50,797,137,096	0.395362
Two pair	18,549,991,152	0.144377
Three of a kind	3,310,535,196	0.025766
Straight	25,219,094,136	0.196284
Flush	3,229,836,828	0.025138
Full house	975,510,000	0.007593
Four of a kind	43,198,800	0.000336
Straight flush	98,961,348	0.000770
Royal flush	9,485,064	0.000074
Total	128,482,615,800	1.000000

Table 6 shows the number of combinations for each hand of a second player, given that the first player has a flush.

Table 6 — First Player has a Flush

Event	Pays	Probability
Less than pair	13,060,678,788	0.155206
Pair	30,076,271,352	0.357410
Two pair	14,200,694,712	0.168754
Three of a kind	2,606,403,900	0.030973
Straight	3,229,836,828	0.038382
Flush	19,608,838,592	0.233021
Full house	1,102,206,960	0.013098
Four of a kind	50,221,200	0.000597
Straight flush	191,762,164	0.002279
Royal flush	23,604,264	0.000281
Total	84,150,518,760	1.000000

Table 7 shows the number of combinations for each hand of a second player, given that the first player has a full house.

Table 7 — First Player has a Full House

Event	Pays	Probability
Less than pair	-	0.000000
Pair	15,829,506,000	0.219222
Two pair	28,751,944,680	0.398185
Three of a kind	12,910,316,760	0.178795
Straight	975,510,000	0.013510
Flush	1,102,206,960	0.015264
Full house	11,661,414,336	0.161499
Four of a kind	966,835,584	0.013390
Straight flush	8,767,440	0.000121
Royal flush	993,600	0.000014
Total	72,207,495,360	1.000000

Table 8 shows the number of combinations for each hand of a second player, given that the first player has a four of a kind.

Table 8 — First Player has a Four of a Kind

Event	Pays	Probability
Less than pair	-	0.000000
Pair	586,278,000	0.125418
Two pair	653,378,400	0.139772
Three of a kind	1,705,867,680	0.364923
Straight	43,198,800	0.009241
Flush	50,221,200	0.010743
Full house	966,835,584	0.206828
Four of a kind	668,375,136	0.142980
Straight flush	390,960	0.000084
Royal flush	44,160	0.000009
Total	4,674,589,920	1.000000

Table 9 shows the number of combinations for each hand of a second player, given that the first player has a straight flush.

Table 9 — First Player has a Straight Flush

Event	Pays	Probability
Less than pair	85,751,460	0.110699
Pair	214,250,184	0.276582
Two pair	109,829,304	0.141782
Three of a kind	19,970,844	0.025781
Straight	98,961,348	0.127752
Flush	191,762,164	0.247552
Full house	8,767,440	0.011318
Four of a kind	390,960	0.000505
Straight flush	44,354,840	0.057259
Royal flush	596,856	0.000770
Total	774,635,400	1.000000

Table 10 shows the number of combinations for each hand of a second player, given that the first player has a royal flush.

Table 10 — First Player has a Royal Flush

Event	Pays	Probability
Less than pair	10,532,592	0.117164
Pair	25,380,864	0.282336
Two pair	12,673,584	0.140981
Three of a kind	2,304,216	0.025632
Straight	9,485,064	0.105512
Flush	23,604,264	0.262573
Full house	993,600	0.011053
Four of a kind	44,160	0.000491
Straight flush	596,856	0.006639
Royal flush	4,280,760	0.047619
Total	89,895,960	1.000000

The following table shows the number of combinations for each hand of player 1 by the winner of the hand.

Table 11 — Winning Player by Hand of Player 1 — Combinations

Player 1	Win	Tie	Loss
Less than pair	76,626,795,600	11,681,317,560	395,983,710,240	484,291,823,400
Pair	496,857,988,764	38,757,694,752	683,256,278,484	1,218,871,962,000
Two pair	419,896,266,012	34,054,545,168	199,549,574,820	653,500,386,000
Three of a kind	97,664,829,948	4,647,370,128	32,024,879,724	134,337,079,800
Straight	103,685,076,072	15,662,001,240	9,135,538,488	128,482,615,800
Flush	71,523,195,288	2,910,219,176	9,717,104,296	84,150,518,760
Full house	62,810,500,464	5,179,382,208	4,217,612,688	72,207,495,360
Four of a kind	4,240,864,800	198,204,864	235,520,256	4,674,589,920
Straight flush	734,237,144	35,247,960	5,150,296	774,635,400
Royal flush	85,615,200	4,280,760	-	89,895,960
Total	1,334,125,369,292	113,130,263,816	1,334,125,369,292	2,781,381,002,400

The following table shows the probability for each hand of player 1 by the winner of the hand. The bottom row shows that each player has a 47.97% chance of winning and a 4.07% chance of a tie.

Table 12 — Winning Player by Hand of Player 1 — Probabilities

Player 1 Hand	Player 1	Tie	Player 2	Total
Less than pair	0.027550	0.004200	0.142369	0.174119
Pair	0.178637	0.013935	0.245654	0.438225
Two pair	0.150967	0.012244	0.071745	0.234955
Three of a kind	0.035114	0.001671	0.011514	0.048299
Straight	0.037278	0.005631	0.003285	0.046194
Flush	0.025715	0.001046	0.003494	0.030255
Full house	0.022582	0.001862	0.001516	0.025961
Four of a kind	0.001525	0.000071	0.000085	0.001681
Straight flush	0.000264	0.000013	0.000002	0.000279
Royal flush	0.000031	0.000002	0.000000	0.000032
Total	0.479663	0.040674	0.479663	1.000000

Written by: Michael Shackleford

The following Texas Holdem odds table highlights some common probabilities that you may encounter in Hold'em. It is not vital that you learn these probabilities, but it is useful to be aware of the chances of certain situations arising.

Texas Hold'em odds chart.

Situation	Percentage Odds	Ratio Odds
Preflop Probabilities:
Dealt AA.	0.45%	220 to 1
Dealt AK.	1.2%	82 to 1
Dealt AKs.	0.3%	331 to 1
Dealt 72o.	0.9%	109 to 1
Being dealt AA vs. KK (heads up).	0.004%	22,559 to 1
Dealt a pocket pair.	6%	16 to 1
Dealt suited connectors.	4%	24 to 1
Flop Probabilities:
Flopping a pair.	32.4%	2.2 to 1
Flopping a set (with pockets).	11.8%	7.5 to 1
Paired Board:
2 players, probability of trips.	17%	4.8 to 1
3 players, probability of trips.	26%	3 to 1
4 players, probability of trips.	34%	2 to 1
5 players, probability of trips.	43%	1.4 to 1

How to use Texas Hold'em odds.

The odds in this Texas Hold'em odds table are unlikely to directly help your overall strategy, but they are pretty interesting nonetheless. The Texas Hold'em odds for each of the different situations have been given in both percentage and ratio odds, so use whichever format you feel comfortable with.

Other poker odds charts.

For more useful odds charts that you can use for when you are working out whether or not to call when on a drawing hand, use the following tables: