Poker Full House Probability
Introduction
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. Find the probability of getting A full house in poker consists of three of a kind, in a five card hand, if 5 cards are dealt at random from a standard deck of 52.-Comment: I think a full house is 3 of one type of card and 2 of another type. Pick a type of card: 13 ways Pick 3 of that type of card: 4C3=4 ways Pick a different type of card. Probability of a Full House We follow a similar process to find the probability of a full house. First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this in the previous section, and found that there are 2,598,960 distinct poker hands. The probability of getting Full House is calculated as below. There are 2,598,960 unique poker hands. Of those, 3,744 are Full House. Therefore, the probability of being dealt full house is: P(full house) = 3744 / 2,598,960 =1.441⋅10−3 or 1 in 694 In stud poker, players get Full House about one time in every 694 deals.
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 |
Poker Full House Probability Rules
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 |