Poker Odds


Learning to calculate poker odds can be confusing for a poker novice. There is a very basic and practical Odds for Dummies at Cardschat that can help get you started before you delve into the mathematics below. As overwhelming as it may seem if you really want to take your poker game to a new level this is something you will need to study and get your arms around. Your decisions need to go beyond reads on your opponents, position and the cards you are dealt.

Combinations


  • Different flop combinations = 19,600

    • MATH: (50*49*48)/(3*2*1)

    • EXPLANATION: This is when you are dealt 2 cards and want to know how many flop combinations there are with 50 cards left. There is a 1 in 50 chance of any particular card being the first flop card. There is a 1 in 49 chance of any particular card being the second flop card and a 1 in 48 chance of any particular card being the third flop card. Then we divide by 6 is because the order of the cards on the flop doesn't matter so any flops that have the same cards but in different order will be considered "duplicate" flops and will not be considered. There are six different ways that a set of 3 cards can be ordered. Either of the 3 cards can be the first card. Then only 2 cards can be the 2nd card and only 1 card can be the last card (3*2*1=6).

  • Different board combinations = 2,118,760

    • MATH: (50*49*48*47*46)/(5*4*3*2*1)

    • EXPLANATION: This is when you are dealt 2 cards and want to know how many board combinations there are with 50 cards left. Following up on the logic from the last example, there are (50+49+48+47+46) different board combinations but like the last example we want to ignore duplicate combinations where you have the same cards but in different order.

Pre-flop


The probability of being dealt:

  • Pocket aces = 220-to-1 (0.45%)

    • MATH: (52/4)*(51/3)

    • EXPLANATION: The odds of the first card being dealt to you being an Ace are 4 in 52. The odds of your second card being an Ace (given the fact that your first card was an Ace) are 3 in 51.

  • Either pocket aces or pocket kings = 109.5-to-1 (0.91%)

    • or
    • MATH: (52/8)*(51/3)

    • EXPLANATION: The odds of the first card being dealt to you being an Ace or a King are 8 in 52 (4 Aces and 4 Kings). The odds of your second card being the same as your first card are 3 in 51.

  • Any pocket pair = 16-to-1 (5.88%)

    • MATH: (51/3)

    • EXPLANATION: The key to the equation here is to realize that to calculate the odds of getting ANY pocket pair is that it doesn't matter what your first card is. It just matters that your second card matches your first. After getting your first card, there are only 3 cards out of the 51 left that will match your first card.

  • AK suited = 330.5-to-1 (0.3%)

    • MATH: (52/8)*(51/1)

    • EXPLANATION: The odds of your first card being an Ace or King is 8 in 52. Then, there is only one card out of the 51 cards left in the deck that can make you a suited big slick.

  • AK offsuit = 109.5-to-1 (0.9%)

    • MATH: (52/8)*(51/3)

    • EXPLANATION: The odds of your first card being an Ace of a King are 8 in 52. The odds of your other card are 3 out of 51.

  • AK suited or offsuit = 81.88-to-1 (1.21%)

    • or
    • MATH: (52/8)*(51/4)

    • EXPLANATION: The odds of your first card being an Ace of a King are 8 in 52. The odds of your other card are 4 out of 51.

  • Any two suited cards = 3.25-to-1 (23.53%)

    • MATH: (51/12)

    • EXPLANATION: It doesn't matter what your first card is. After you get your first card, there will only be 12 cards out of the 51 cards left that will be the same suit as your first.

  • Either pocket aces, pocket kings or AK = 46.36-to-1 (2.11%)

    • oror
    • MATH: (52/8)*(51/7)

    • EXPLANATION: This is a useful calculation to know if you are getting deep into a tournament and need to double up and want to wait for a really good hand. There are 8 Aces or Kings you can get as your first card. After that, there are 7 Aces or Kings left out of 51 cards.

Pocket Pairs


  • Flopping a set = 8.28-to-1 (10.8%)

    •  

      HandFlop

    • MATH: ((2/50)*(48/49)*(44/48))*3

    • EXPLANATION: A card that matches your set has a 2 out of 50 chance of coming out. The next card can be any card other than a card that will make you quads, which is 48 out of the remaining 49 cards. The 3rd flop card can be any card other than a card that gives you quads OR that matches the 2nd flop card to give you a full house, which is 44 out of 48 remaining cards. Then you multiply by 3 to take into account the 3 different ways that hitting the set can happen (hitting your set on the 1st, 2nd, or 3rd flop card). As a side note, there is a common misperception that the odds of flopping a set are 7.5-to-1. Technically, this is not true because this calculation includes the odds of hitting a full house with your set (which is no longer a set).

  • Flopping a full house = 135.1-to-1 (0.73%)

    •  

      HandFlop

    • MATH: ((2/50)*(48/49)*(3/48))*3

    • EXPLANATION: A card that matches your set has a 2 out of 50 chance of coming out. The next card can be any card other than a card that will make you quads (48 out 49 cards). The 3rd flop card has to match the 2nd flop card, (3 out of 48 cards). Then you multiply by 3 to take into account the 3 different ways that you can hit the full house. It should be noted that we are not including the probability of the flop being all the same cards, which gives you a full house without hitting the flop.

  • Flopping quads = 407.33-to-1 (0.245%)

    •  

      HandFlop

    • MATH: (2/50)*(1/49)*(48/48) + (48/50)*(2/49)*(1/48) + (2/50)*(48/49)*(1/48)

    • EXPLANATION: The first flop card has to be the same as your pocket pair (2 out of 50). The second flop card has to be the same as your pocket pair (1 out of 49). The third card can be any other card (48 out of 48). Then we multiply by 3 since there are 3 different combinations of ways to hit the quads.

  • Flopping a set or better = 7.51-to-1 (11.76%)

    • MATH: 1-((48/50)*(47/49)*(46/48))

    • EXPLANATION: Sometimes it's easier to calculate the odds of the opposite. Here, I calculate the odds of NOT flopping a set and subtract it from 100%. To calculate the odds of not flopping a set, all the cards on the flop can't have the same rank as your pocket pair. For the first card there are 48 out of 50 cards that won't hit your set. For the second card there are 47 out of 49 cards and for the third card there are 46 out of 48 cards that won't hit your set. This probability gives you the oft-quoted 7.5-to-1 odds.

With 2 suited cards


  • Flopping a flush = 0.84% (117.79-to-1)

    •  

      HandFlop

    • MATH: (50/11)*(49/10)*(48/9)

    • EXPLANATION: There are 13 cards of each suit. Assuming you have 2 suited cards then there are only 11 cards left of that suit and only 50 total cards left in the deck. The probability of the first flop card being of your suit is 11 in 50. Then the second card 10 in 49 and then the last card 9 in 48.

  • Flopping a flush draw = 8.14-to-1 (10.9%)

    •  

      HandFlop

    • MATH: (11/50)*(10/49)*(39/48)*3

    • EXPLANATION: There is a 11-in-50 chance of the first flop card being a flush card and a 10-in-49 chance of the 2nd flop card being a flush card. Then there is a 39-in-48 chance of the third flop card NOT being a flush card. Then we multiply by 3 because there are 3 different possible flops with 2 of the flush cards.

  • Flopping a backdoor flush draw (3 flush cards) = 1.4-to-1 (41.6%)

    •  

      HandFlop

    • MATH: (11/50)*(39/49)*(38/48)*3

    • EXPLANATION: There is a 11-in-50 chance of the first flop card being a flush card. There is a 39-in-49 chance of the 2nd flop card NOT being a flush card. Then there is a 38-in-48 chance of the third flop card NOT being a flush card. Then we multiply by 3 to adjust for the number of possible different flops where there is only 1 flush card.

  • Making a flush by the river = 14.6-to-1 (6.4%)

    •  

      HandBoard

    • MATH: ((COMBIN(11,3)*COMBIN(39,2)) + (COMBIN(11,4)*COMBIN(39,1)) + (COMBIN(11,5)))/((50*49*48*47*46)/(5*4*3*2*1))

    • EXPLANATION: We add up the 3 possible mutually-exclusive scenarios - having the board show 3 of the flush suit, 4 of the flush suit, or 5 of the flush suit. Then we divide this sum by the total number of possible board combinations. Note: some of the flushes may be straight flushes.

With 2 unsuited cards


  • Flopping a flush draw = 43.55-to-1 (2.2%)

    •  

      HandFlop

    • MATH: (50/24)*(49/11)*(48/10)

    • EXPLANATION: There are only 12 cards left of each suit that you hold. The probability of the first flop card being of either of your suits is 24 in 50. The chances that the second flop card is the same suit as the first flop card is 11 in 49. The chance that the third flop card is the same suit as the other flop cards is 10 in 48.

  • Flopping a backdoor flush draw = 2.91-to-1 (25.6%)

    •  

      HandFlop

    • MATH: (24/50)*(11/49)*(38/48)*3

    • EXPLANATION: There are only 12 cards left of each suit that you hold. The probability of the first flop card being of either of your suits is 24 in 50. The chances that the second flop card is the same suit as the first flop card is 11 in 49. The chance that the third flop card NOT being the same suit as the other flop cards is 38 in 48. We then multiply by 3 because there are 3 unique combinations of flops can contain 2 out of 3 of your suit.

  • Making a flush by the river = 78.46-to-1 (1.26%)

    •  

      HandBoard

    • MATH: (((COMBIN(11,4)*COMBIN(39,1))+(COMBIN(11,5)))*2)/((50*49*48*47*46)/(5*4*3*2*1))

    • EXPLANATION: We add up the 2 possible mutually-exclusive scenarios where the board shows 4 of the flush suit or 5 of the flush suit. Then we multiply that by 2 since we are holding 2 different suits which can fop a flush draw. Then we divide this sum by the total number of possible board combinations.

With 2 non-pair cards


  • Flopping at least a pair (using your pocket cards) = 2.1-to-1 (32.4%)

    •  

      HandFlop

    • MATH: 1-((44/50)*(43/49)*(42/48))

    • EXPLANATION: Without possible straights and flushes, the only possible way to flop a pair or better is to have the flop hit your hand. So, I calculated the odds of the flop missing you and subtracting it from 100%. The odds of each flop card missing your pocket cards are 44-in-50, 43-in-49, and 42-in-48, respectively.

  • Exactly one pair (using your pocket cards) = 2.45-to-1 (29.0%)

    •  

      HandFlop

    • MATH: ((6/50)*(44/49)*(43/48))*3

    • EXPLANATION: The chances of the first flop card hitting you is 6-in-50. The chances of the second and third cards missing you are 44-in-49 and 43-in-48, respectively. Then we multiply by 3 because there are 3 unique flop combinations where exactly 1 card matches the pocket cards.

  • Two pair (using both of your pocket cards) = 48.49-to-1 (2.02%)

    •  

      HandFlop

    • MATH: ((6/50)*(3/49)*(44/48))*3

    • EXPLANATION: The chances of the first flop card hitting either of your pocket cards is 6-in-50. The chances of the second hitting your OTHER pocket card is 3-in-49. The chances of the third flop card missing you is 44-in-48. Then we multiply by 3 because there are 3 unique flop combinations where you can flop 2 pair using both of your pocket cards.

  • Trips = 73.24-to-1 (1.3%)

    •  

      HandFlop

    • MATH: ((6/50)*(2/49)*(44/48))*3

    • EXPLANATION: The chances of the first flop card hitting either of your pocket cards is 6-in-50. The chances that the second flop card will match the first is 2-in-49. The chances that the third flop card will miss your pocket card sis 44-in-48. We multiply by 3 because there are 3 unique flop combinations where you can flop trips.

  • A full house = 1,088-to-1 (.09%)

    •  

      HandFlop

    • MATH: (3/50)*(2/49)*(3/48)*6

    • EXPLANATION: The first card has to match one of your hole cards (3 out of 50). The second has to make you trips (2 out of 49). And the third has to match your other card (3 out of 48). Then we multiply by 6 because there are 6 different combinations of flops to flop the full house.

  • Quads = 9,799-to-1 (.01%)

    •  

      HandFlop

    • MATH: (50/6)*(49/2)*(48/1)

    • EXPLANATION: The first flop card has to be of the same rank as one of the 2 cards you hold (6 cards out of 50). The second flop card has to be the same as the first flop card (2 out of 49). The last flop card is the only card left out of 48 of the same rank as the other two flop cards.

Odds that the flop will be:


  • Three of a kind = 424-to-1 (0.24%)

    • Flop

    • MATH: (51/3)*(50/2)

    • EXPLANATION: Your pocket cards are ignored so we start with 52 cards. It doesn't matter what the first flop card is - only that the second and third flop cards have the same rank as the first. After the first card comes on the flop there are only 3 cards of that rank left and the odds of the second flop card being the same rank as the first are 3 in 51. Then the odds of the third flop card being the same rank as the first two are 2 in 50.

  • A pair = 4.9-to-1 (16.94%)

    • Flop

    • MATH: (52/52)*(3/51)*(48/50)*3

    • EXPLANATION: The first card can be any card, the second card has to match the first, and the third card can be any card other than the 2 cards that match the first two. Then we multiply by 3 because there are 3 ways for the flop to pair.

  • Three suited cards = 18.3-to-1 (5.18%)

    • Flop

    • MATH: (51/12)*(50/11)

    • EXPLANATION: Your pocket cards are ignored so we start with 52 cards. It doesn't matter what the first flop card is - only that the second and third flop cards have the same suit as the first card. After the first card is dealt there is only 12 cards left of that suit. After the turn there are only 11 cards left of that suit.

  • Two suited cards = .82-to-1 (55.1%)

    • Flop

    • MATH: (52/52)*(12/51)*(39/50)*3

    • EXPLANATION: The first card can be any card, the second card's suit has to match the first, and the third card's suit can be any suit other than the 2 cards that match the first two. Then we multiply by 3 because there are 3 ways for the flop to pair.

  • No suited cards = 1.5-to-1 (39.8%)

    • Flop

    • MATH: (52/52)*(39/51)*(26/50)

    • EXPLANATION: Your pocket cards are ignored so we start with 52 cards. It doesn't matter what the first card on the flop is - only that the other two flop cards don't match the suit of the others. When the second card comes it has to be one of the 3 other suits (13 cards of each suit) than the first flop card. So that's 39 cards out of 51. Then the third flop card needs to be one of the 2 suits that hasn't been flopped yet. So that is 26 cards out of 50.

  • Three cards in sequence = 28-to-1 (3.5%)

    • Flop

  • Two cards in sequence = 1.5-to-1 (40%)

    • Flop

  • No cards in sequence = .8-to-1 (56%)

    • Flop

The turn


  • A full house or better from a set on the next card = 5.7-to-1 (14.89%)

    •  

      HandFlop

    • MATH: (47/7)

    • EXPLANATION: There are 6 cards to make a full house and 1 card to make you quads.

  • A full house from two pair on the next card = 10.8-to-1 (8.51%)

    •  

      HandFlop

    • MATH: (47/4)

    • EXPLANATION: You have 4 cards to hit your boat.

  • A flush from a four-flush on the next card = 4.22-to-1 (19.15%)

    •  

      HandFlop

    • MATH: (47/9)

    • EXPLANATION: You have 9 cards left of your suit to hit your flush.

  • A straight from an open-ended straight draw on the next card = 4.9-to-1 (17.02%)

    •  

      HandFlop

    • MATH: (47/8)

    • EXPLANATION: You have 8 cards left to hit your straight.

  • A straight from a gutshot straight draw on the next card = 10.8-to-1 (8.51%)

    •  

      HandFlop

    • MATH: (47/4)

    • EXPLANATION: You have 4 cards left to hit your straight.

  • A pair from two overcards = 6.8-to-1 (12.77%)

    •  

      HandFlop

    • MATH: (47/6)

    • EXPLANATION: You have 6 cards to hit your pair.

Flop to river


  • A full house or better from a set by the river = 1.99-to-1 (33.4%)

    •  

      HandFlop

    • MATH: 1-((40/47)*(36/46))

    • EXPLANATION: There are 10 cards on the turn that will make you a full house. Assuming that you don't hit it on the turn, the turn card will add 3 more cards for you to hit your boat on the river.

  • A full house or better from two pair by the river = 5.07-to-1 (16.5%)

    •  

      HandFlop

    • MATH: 1-((43/47)*(42/46))

    • EXPLANATION: You have four outs to hit your full house on either the turn on river.

  • A flush from a four-flush by the river = 1.86-to-1 (35.0%)

    •  

      HandFlop

    • MATH: 1-((40/47)*(36/46))

    • EXPLANATION: You have to hit one of the 7 flush cards either on the turn OR the river.

  • A backdoor flush by the river = 23.02-to-1 (4.16%)

    •  

      HandFlop

    • MATH: (10/47)*(9/46)

    • EXPLANATION: You have to hit one out of the 10 flush cards on the turn and hit one out of the 9 flush cards on the river.

  • A straight from an open-ended straight draw by the river = 2.18-to-1 (31.45%)

    •  

      HandFlop

    • MATH: 1-((39/47)*(38/46))

    • EXPLANATION: You have to hit one out of the 8 straight cards either on the turn or the river.

  • A straight from a gutshot straight draw by the river = 5.07-to-1 (16.47%)

    •  

      HandFlop

    • MATH: 1-((43/47)*(42/46))

    • EXPLANATION: You have to hit one out of the 4 straight cards either on the turn or the river.

  • A pair or better from two overcards by the river = 3.14-to-1 (24.14%)

    •  

      HandFlop

    • MATH: 1-((41/47)*(40/46))

    • EXPLANATION: You have to hit one out of the 6 cards that can make you a pair either on the turn or the river.

The river


  • A full house or better from a set = 3.6-to-1 (21.7%)

    •  

      HandBoard

    • MATH: (46/10)

    • EXPLANATION: You have 9 different cards to hit your boat and 1 card to hit quads.

  • A full house from two pair on the final card = 10.5-to-1 (8.7%)

    •  

      HandBoard

    • MATH: (46/4)

    • EXPLANATION: You have 4 cards left that match your two pair.

  • A flush from a four-flush on the final card = 4.11-to-1 (19.6%)

    •  

      HandBoard

    • MATH: (46/9)

    • EXPLANATION: You have 9 cards to hit your pair.

  • An open-ended straight draw = 4.75-to-1 (17.4%)

    •  

      HandBoard

    • MATH: (46/8)

    • EXPLANATION: You have 8 cards to hit your straight.

  • A gutshot straight draw = 10.5-to-1 (8.7%)

    •  

      HandBoard

    • MATH: (46/4)

    • EXPLANATION: You have 4 cards to hit your straight.

  • A pair from two overcards = 6.67-to-1 (13.0%)

    •  

      HandBoard

    • MATH: (46/6)

    • EXPLANATION: You have 8 cards to hit your straight.

Hand matchups - Pre-flop


  • Overpair vs. Underpair:

    • vs.

      80.9%19.1%

    • vs.

      80.2%19.8%

  • Pocket Aces vs. non-pair:

    • vs.

      86.3%13.6%

    • vs.

      80.1%19.3%

  • Pocket Aces vs. Suited Connectors:

    • vs.

      77.0%23.0%

  • Pair vs. Two Overcards

    • vs.

      56.9%43.1%

    • vs.

      51.6%48.3%

  • Pair vs. One Overcard (unsuited)

    • vs.

      70.3%29.7%

  • Pair vs. One Overcard (suited)

    • vs.

      66.6%33.8%

  • Dominated Hands

    • vs.

      73.7%26.3%

    • vs.

      69.4%30.6%

    • vs.

      72.6%27.4%

  • Two Overcards vs. Non-Pair

    • vs.

      62.6%37.4%

    • vs.

      58.8%41.2%

  • One overcard vs. Non-pair

    • vs.

      56.2%43.9%

    • vs.

      52.6%47.4%

Hand matchups - On the flop


  • Flush draw vs. top pair

    •   

      38.6%Flop61.4%

  • Flush draw vs. 2 pair

    •   

      35.9%Flop64.1%

  • Flush draw vs. trips

    •   

      25.6%Flop74.4%

  • Flush vs. top pair

    •   

      97.1%Flop2.9%

  • Flush vs. 2 pair

    •   

      82.5%Flop17.5%

  • Flush vs. a set

    •   

      65.5%Flop34.5%

  • Bottom 2 pair vs. top pair

    •   

      74.5%Flop25.5%

  • Bottom 2 pair vs. an overset

    •   

      0.2%Flop99.8%

  • Top 2 pair vs. an underset

    •   

      16.8%Flop83.2%

  • Top pair(bad kicker) vs. top pair (good kicker)

    •   

      12.7%Flop87.3%

  • Open-ended straight draw vs. a set

    •   

      25.9%Flop74.1%

  • Open-ended straight draw AND a flush draw vs. top pair

    •   

      55.3%Flop44.6%

  • Open-ended straight draw AND a flush draw vs. a set

    •   

      40.6%Flop59.4%

  • A set vs. a higher set

    •   

      4.3%Flop95.7%

  • Inside straight draw vs. overpair

    •   

      18.7%Flop81.3%

  • Inside straight draw vs. underpair

    •   

      38.6%Flop61.4%

  • Inside straight draw vs. Two pair

    •   

      19.7%Flop80.3%

  • Inside straight draw vs. a set

    •   

      13.7%Flop86.2%

  • Underpair vs. 2 overcards

    •   

      75.3%Flop24.7%

Hand matchups - The turn


  • Flush draw vs. top pair

    •   

      20.5%Board79.5%

  • Flush draw vs. 2 pair

    •   

      18.2%Board81.2%

  • Flush draw vs. a set

    •   

      15.9%Board84.1%

  • Flush vs. 2 pair

    •   

      90.1%Board9.0%

  • Flush vs. a set

    •   

      77.3%Board22.7%

  • Bottom 2 pair vs. top pair

    •   

      81.8%Board18.2%

  • Top pair(bad kicker) vs. top pair (good kicker)

    •   

      6.8%Board93.1%

  • Open-ended straight draw vs. a set

    •   

      18.2%Board81.2%

  • Open-ended straight draw AND a flush draw vs. top pair

    •   

      34.1%Board65.9%

  • Open-ended straight draw AND a flush draw vs. a set

    •   

      29.5%Board70.5%

  • A set vs. a higher set

    •   

      2.3%Board97.7%

  • Inside straight draw vs. overpair

    •   

      9.1%Board90.1%

  • Inside straight draw straight draw vs. underpair

    •   

      22.7%Board77.3%

  • Inside straight draw straight draw vs. two pair

    •   

      9.1%Board90.9%

  • Inside straight draw straight draw vs. a set

    •   

      9.1%Board90.1%

COMMENTS:

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The probability of at least or exactly one pair or trips when holding non-paired pocket cards that are given above are different from the probabilities given by the calculator in the link below - who is this so?

http://www.texasholdem-poker.com/cal...age&decimals=2

njaalgw@gmail.com on November 27, 2013

Odds of.Quads on turn v full house full of lower cards? (guess this is the same as under set v higher set)

NB5 on September 13, 2013

Is there a way to get this guide in a PDF format? Its just great

Latamgrinder on August 20, 2013

Poker Odds

HPG ADMIN on March 5, 2013

Jump Links

Combinations
Pre-flop
Pocket pairs
2 suited cards
2 unsuited cards
2 non-paired cards
Flop odds
The turn
Flop to river
The river
Hands - pre-flop
Hands - the flop
Hands - the turn