A “poker hand” consists of 5 unordered cards from a standard deck of 52. There are (52 5) = 2, 598, 960 possible poker hands. Let us now calculate the probability of each of the standard kinds of poker hands. This hand consists of values 10, J, Q, K, A, all of the same suit. There are royal flushes, for a probability of 0. I'm dealing with an exercise which deals with the poker game. I need to calculate the probability of getting a full house. Full house is getting 3 cards of the same type and 2 cards of the same type. I've made a research, but I cannot understand why the combination for getting a full house is.

1. Poker Dice Full House Probability
2. Poker Full House Probability Rules
3. Poker Full House Probability Game

1. The probability is 0.021128.
2. 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.

The main underpinning of poker is math – it is essential. For every decision you make, while factors such as psychology have a part to play, math is the key element.

In this lesson we’re going to give an overview of probability and how it relates to poker. This will include the probability of being dealt certain hands and how often they’re likely to win. We’ll also cover how to calculating your odds and outs, in addition to introducing you to the concept of pot odds. And finally we’ll take a look at how an understanding of the math will help you to remain emotional stable at the poker table and why you should focus on decisions, not results.

What is Probability?

Probability is the branch of mathematics that deals with the likelihood that one outcome or another will occur. For instance, a coin flip has two possible outcomes: heads or tails. The probability that a flipped coin will land heads is 50% (one outcome out of the two); the same goes for tails.

Probability and Cards

When dealing with a deck of cards the number of possible outcomes is clearly much greater than the coin example. Each poker deck has fifty-two cards, each designated by one of four suits (clubs, diamonds, hearts and spades) and one of thirteen ranks (the numbers two through ten, Jack, Queen, King, and Ace). Therefore, the odds of getting any Ace as your first card are 1 in 13 (7.7%), while the odds of getting any spade as your first card are 1 in 4 (25%).

Unlike coins, cards are said to have “memory”: every card dealt changes the makeup of the deck. For example, if you receive an Ace as your first card, only three other Aces are left among the remaining fifty-one cards. Therefore, the odds of receiving another Ace are 3 in 51 (5.9%), much less than the odds were before you received the first Ace.

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Pre-flop Probabilities: Pocket Pairs

In order to find the odds of getting dealt a pair of Aces, we multiply the probabilities of receiving each card:

(4/52) x (3/51) = (12/2652) = (1/221) ≈ 0.45%.

To put this in perspective, if you’re playing poker at your local casino and are dealt 30 hands per hour, you can expect to receive pocket Aces an average of once every 7.5 hours.

The odds of receiving any of the thirteen possible pocket pairs (twos up to Aces) is:

(13/221) = (1/17) ≈ 5.9%.

In contrast, you can expect to receive any pocket pair once every 35 minutes on average.

Pre-Flop Probabilities: Hand vs. Hand

Players don’t play poker in a vacuum; each player’s hand must measure up against his opponent’s, especially if a player goes all-in before the flop.

Here are some sample probabilities for most pre-flop situations:

Post-Flop Probabilities: Improving Your Hand

Now let’s look at the chances of certain events occurring when playing certain starting hands. The following table lists some interesting and valuable hold’em math:

Many beginners to poker overvalue certain starting hands, such as suited cards. As you can see, suited cards don’t make flushes very often. Likewise, pairs only make a set on the flop 12% of the time, which is why small pairs are not always profitable.

PDF Chart

We have created a poker math and probability PDF chart (link opens in a new window) which lists a variety of probabilities and odds for many of the common events in Texas hold ‘em. This chart includes the two tables above in addition to various starting hand probabilities and common pre-flop match-ups. You’ll need to have Adobe Acrobat installed to be able to view the chart, but this is freely installed on most computers by default. We recommend you print the chart and use it as a source of reference.

Odds and Outs

If you do see a flop, you will also need to know what the odds are of either you or your opponent improving a hand. In poker terminology, an “out” is any card that will improve a player’s hand after the flop.

One common occurrence is when a player holds two suited cards and two cards of the same suit appear on the flop. The player has four cards to a flush and needs one of the remaining nine cards of that suit to complete the hand. In the case of a “four-flush”, the player has nine “outs” to make his flush.

A useful shortcut to calculating the odds of completing a hand from a number of outs is the “rule of four and two”. The player counts the number of cards that will improve his hand, and then multiplies that number by four to calculate his probability of catching that card on either the turn or the river. If the player misses his draw on the turn, he multiplies his outs by two to find his probability of filling his hand on the river.

In the example of the four-flush, the player’s probability of filling the flush is approximately 36% after the flop (9 outs x 4) and 18% after the turn (9 outs x 2).

Pot Odds

Another important concept in calculating odds and probabilities is pot odds. Pot odds are the proportion of the next bet in relation to the size of the pot.

For instance, if the pot is $90 and the player must call a $10 bet to continue playing the hand, he is getting 9 to 1 (90 to 10) pot odds. If he calls, the new pot is now $100 and his $10 call makes up 10% of the new pot.

Experienced players compare the pot odds to the odds of improving their hand. If the pot odds are higher than the odds of improving the hand, the expert player will call the bet; if not, the player will fold. This calculation ties into the concept of expected value, which we will explore in a later lesson.

Bad Beats

A “bad beat” happens when a player completes a hand that started out with a very low probability of success. Experts in probability understand the idea that, just because an event is highly unlikely, the low likelihood does not make it completely impossible.

A measure of a player’s experience and maturity is how he handles bad beats. In fact, many experienced poker players subscribe to the idea that bad beats are the reason that many inferior players stay in the game. Bad poker players often mistake their good fortune for skill and continue to make the same mistakes, which the more capable players use against them.

Decisions, Not Results

One of the most important reasons that novice players should understand how probability functions at the poker table is so that they can make the best decisions during a hand. While fluctuations in probability (luck) will happen from hand to hand, the best poker players understand that skill, discipline and patience are the keys to success at the tables.

A big part of strong decision making is understanding how often you should be betting, raising, and applying pressure.
The good news is that there is a simple system, with powerful shortcuts & rules, that you can begin using this week. Rooted in GTO, but simplified so that you can implement it at the tables, The One Percent gives you the ultimate gameplan.

This 7+ hour course gives you applicable rules for continuation betting, barreling, raising, and easy ratios so that you ALWAYS have the right number of bluffing combos. Take the guesswork out of your strategy, and begin playing like the top-1%.

Conclusion

A strong knowledge of poker math and probabilities will help you adjust your strategies and tactics during the game, as well as giving you reasonable expectations of potential outcomes and the emotional stability to keep playing intelligent, aggressive poker.

Remember that the foundation upon which to build an imposing knowledge of hold’em starts and ends with the math. I’ll end this lesson by simply saying…. the math is essential.

By Gerald Hanks

Gerald Hanks is from Houston Texas, and has been playing poker since 2002. He has played cash games and no-limit hold’em tournaments at live venues all over the United States.

Introduction

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

Poker Dice Full House Probability

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.

Poker Full House Probability Rules

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

Poker Full House Probability Game

Written by: Michael Shackleford