Introduction
In a previous post, I looked at the likelihood of winning with the various kinds of hold'em poker hands one could be dealt. For simplicity, I ignored ties. Having had some time to develop my code, I now wish to add tie breaking into the analysis, along with other new insights gathered from capturing more detailed data. The result will be a more refined set of observations useful for winning at hold'em.Where does tie breaking matter? Let's take a look at the chart of Wins, Ties, and Losses against one other player without tie breaking implemented. This consists of 100,000 simulations.
Player1 | Lose | Tie | Win | Lose/All | Tie/All | Win/All |
1.Royal Flush | 0 | 0 | 23 | 0.00 | 0.00 | 1.00 |
2. Straight Flush | 0 | 5 | 140 | 0.00 | 0.03 | 0.97 |
3. Four of a Kind | 0 | 24 | 150 | 0.00 | 0.14 | 0.86 |
4. Full House | 40 | 409 | 2193 | 0.02 | 0.15 | 0.83 |
5. Flush | 80 | 620 | 2121 | 0.03 | 0.22 | 0.75 |
6. Straight | 150 | 768 | 3050 | 0.04 | 0.19 | 0.77 |
7. Three of a Kind | 733 | 1001 | 3184 | 0.15 | 0.20 | 0.65 |
8. Two Pair | 3274 | 8857 | 11148 | 0.14 | 0.38 | 0.48 |
9. One Pair | 14830 | 20968 | 8441 | 0.34 | 0.47 | 0.19 |
10. High Card | 11655 | 6136 | 0 | 0.66 | 0.34 | 0.00 |
Against one player, tie breaking does not seem to matter as much for a three of a kind or better. These hands win most of the time (greater than 50%), so if one always played when one had one of these hands, one would win on average. Similarly, tie breaking does not matter much for a single high card hand since this hand loses most of the time.
Tie breaking does seem to matter for one pair and two pair, however. Tying is very common for both, and no result (win, lose, tie) occurs more than 50% of the time. So getting a more refined look at one pair and two pair hands should provide better guidance for what to do with these sorts of hands.
With Tie Breaking
Now consider the updated table against one other player which includes tie-breaking for all types of hands.Player1 - Tie Breaking | Lose | Tie | Win | Lose/All | Tie/All | Win/All |
1.Royal Flush | 0 | 0 | 23 | 0.00 | 0.00 | 1.00 |
2. Straight Flush | 0 | 4 | 141 | 0.00 | 0.03 | 0.97 |
3. Four of a Kind | 11 | 5 | 158 | 0.06 | 0.03 | 0.91 |
4. Full House | 160 | 182 | 2300 | 0.06 | 0.07 | 0.87 |
5. Flush | 352 | 89 | 2380 | 0.12 | 0.03 | 0.84 |
6. Straight | 150 | 768 | 3050 | 0.04 | 0.19 | 0.77 |
7. Three of a Kind | 1165 | 157 | 3596 | 0.24 | 0.03 | 0.73 |
8. Two Pair | 7079 | 1212 | 14988 | 0.30 | 0.05 | 0.64 |
9. One Pair | 24689 | 1387 | 18163 | 0.56 | 0.03 | 0.41 |
10. High Card | 14552 | 418 | 2821 | 0.82 | 0.02 | 0.16 |
What has changed? Obviously, ties are much more rare. These now only occur when player1 and player2 have the exact same hand (as opposed to the same type of hand, as was previously the case). All hands tie below 7% except the straight, which ties at 19%.
The better hands got better and the worse hands got worse. Everything above and including three of a kind improved in odds of winning. On the other hand, the high card greatly increased in its chances of losing. Of particular importance given the previous discussion, the two pair now wins the majority of the time (64%) while the one pair loses the majority of the time (56%).
Thus, based on this updated information, it would appear that against one other player one should (in general):
- play when one has a two pair or better
- fold when one has a one pair or worse
The Tie-Breaker
In the event of a tie in type of hand, what will break the tie? First, note that 62% of the time, there will not be a tie in the type of hand. However, in 19% of hands, the high card is what breaks the tie. After that, the single one pair breaks the tie at 9%, the first pair in a two pair at 4%, and the tie is not broken at 3%.Tie Breaker | Count |
No tie | 61980 |
High Card | 19394 |
One Pair | 9438 |
First Pair | 3575 |
Tie | 3454 |
Second Pair | 1915 |
Pair | 161 |
Three of a Kind | 83 |
Clearly the high card is very important in the event of a tie in type of hand, breaking about half of all ties. How high of a high card do you need to break the tie?
Tie Breaker - Key Cards: High Card | Lose | Win | Win% |
14 | 0 | 2629 | 100.00 |
13 | 1204 | 2110 | 63.67 |
12 | 1614 | 1543 | 48.88 |
11 | 1601 | 1168 | 42.18 |
10 | 1430 | 846 | 37.17 |
9 | 1192 | 581 | 32.77 |
8 | 973 | 370 | 27.55 |
7 | 700 | 235 | 25.13 |
6 | 470 | 104 | 18.12 |
5 | 326 | 42 | 11.41 |
4 | 131 | 9 | 6.43 |
2 | 61 | 0 | 0.00 |
3 | 55 | 0 | 0.00 |
It looks like a King (13) or better is needed as a high card tie breaker to win more than 50% of the time when depending on this to break the tie. Let's look at the two types of hands where a tie-breaker matters the most: the one pair and the two pair.
One Pair
Consider the table below of Tie Breakers for Player1's one pair hands:Tie Breaker - One Pair | Count | Percentage |
Not a tie | 23271 | 52.6 |
1. One Pair | 9438 | 21.3 |
3. High Card | 5307 | 12.0 |
4. High Card | 3196 | 7.2 |
5. High Card | 1640 | 3.7 |
6. Tie | 1387 | 3.1 |
In most cases, a one pair does not tie (53%). After that, it is the pair itself that breaks the tie at 21%, and the highest single card at 12%. Having a good high card can often prevent a tie or loss when one has a one-pair hand.
What is a good pair to break a tie with? Not surprisingly, the higher the pair, the better the chances of breaking the tie in your favor are. Looking at the table below, we can see that a pair of Aces always wins (722-0) whereas as a pair of 2s always loses (0-785).
Tie Breaker - Key Cards: One Pair | Lose | Win |
14 | 0 | 722 |
13 | 60 | 660 |
12 | 97 | 645 |
11 | 184 | 547 |
10 | 228 | 482 |
9 | 280 | 451 |
8 | 363 | 352 |
7 | 385 | 271 |
6 | 485 | 232 |
5 | 523 | 201 |
4 | 662 | 88 |
3 | 684 | 51 |
2 | 785 | 0 |
Key takeaway:
- One needs to have a pair of 9s or better to win the tie break most of the time.
What about the high card? Again, not surprisingly, the higher the kicker, the better the chances are of winning the tie break. The table below shows that, with an Ace breaking the tie, one always wins. What is odd is the Lose column. What is going on here?
Tie Breaker - First High Card | Lose | Win |
14 | 0 | 1388 |
13 | 607 | 692 |
12 | 708 | 335 |
11 | 585 | 145 |
10 | 393 | 54 |
9 | 225 | 21 |
8 | 109 | 1 |
7 | 32 | 3 |
6 | 9 | 0 |
Key takeaway
- Don't rely on anything less than a King kicker to win a one-pair tie break depending on the kicker.
- To win a one-pair tie break more than 50% of the time, have at least a pair of 9s and a high card of a King or better.
Two Pair
Consider the table below of Tie Breakers for Player1's two pair hands:Tie Breaker - Two Pair | Count | Percentage |
Not a Tie | 14422 | 61.95 |
1. First Pair | 3575 | 15.36 |
3. High Card | 2155 | 9.26 |
2. Second Pair | 1915 | 8.23 |
3. Tie | 1212 | 5.21 |
Most of the time, a two-pair does not tie. However, 15% of the time it does tie and it is the first pair that breaks the tie. Interestingly enough, it is the kicker that breaks the tie next most often at 9%, followed by the second pair at 8%. The kicker is slightly more important than the second pair in breaking ties.
Not surprisingly, the better the first pair, the more likely one will win the tie break.
Tie Breaker - Key Cards: First Pair | Lose | Win |
14 | 0 | 412 |
13 | 69 | 355 |
12 | 125 | 294 |
11 | 171 | 212 |
10 | 205 | 163 |
9 | 225 | 154 |
8 | 246 | 85 |
7 | 218 | 48 |
6 | 191 | 41 |
5 | 161 | 18 |
4 | 116 | 4 |
3 | 62 | 0 |
It seems one needs a first pair of Jacks to have a better than 50% chance of winning the tie break. The lose column is a little difficult to understand at first, but it is combining the likelihood of having a first pair of a certain rank along with the chance of losing. So for example, it is much more unlikely to have a first pair of 3s than a first pair of Aces because Aces can be a first pair to a second pair of Kings, Queens, etc. On the other hand, a first pair of 3s can only have a second pair of 2s. This combination is much less likely to occur. Also, since a first pair of 3s is the lowest possible first pair possible, it is never going to win a tie break with another two pair hand, whereas a first pair of Aces will never lose a tie break with another two pair hand.
Key takeaway:
- A pair of Jacks is needed as a first pair to win a tie break most of the time against another two pair hand.
Tie Breaker - Key Cards: High Card | Lose | Win |
14 | 0 | 335 |
13 | 76 | 293 |
12 | 129 | 171 |
11 | 143 | 131 |
10 | 134 | 81 |
9 | 117 | 60 |
8 | 104 | 30 |
7 | 73 | 10 |
6 | 71 | 7 |
5 | 52 | 6 |
4 | 37 | 1 |
2 | 53 | 0 |
3 | 41 | 0 |
Key takeaway:
- A Queen kicker is needed to win a tie break most of the time against another two pair hand when relying on the kicker to break the tie.
Tie Breaker - Key Cards: Second Pair | Lose | Win | Win% |
13 | 0 | 37 | 100.0 |
12 | 5 | 76 | 93.8 |
11 | 15 | 95 | 86.4 |
10 | 19 | 106 | 84.8 |
9 | 40 | 118 | 74.7 |
8 | 46 | 107 | 69.9 |
7 | 91 | 103 | 53.1 |
6 | 95 | 110 | 53.7 |
5 | 111 | 77 | 41.0 |
4 | 151 | 57 | 27.4 |
3 | 176 | 43 | 19.6 |
2 | 237 | 0 | 0.0 |
Because this is the second pair, it is very unlikely that one will have, for example, a pair of Kings as a second pair, because there is only the possibility of having a pair of Aces for a first pair in that case. In contrast, a second pair of 2s can have 3s, 4s, on up as a first pair, so this is much more likely to be the case. Thus, low second pairs are much more common than high second pairs.
In addition, in most cases, a two-pair hand will require that one of the pairs is in the community cards. Doing an analysis on two pair hands reveals that 48% of the time, both first pair cards are in the community cards, 47% of the time one of the first pair cards is in the community cards, and only 5% of the time will no first pair cards be in the community cards. In two pair ties, this changes to 63% for both, 31% for one, and 5% for no first pair cards in the community cards.
For second pair cards, 53% of the time both cards are in the community, 43% of the time one card is in the community, and 4% of the time no cards are in the community. When looking only at ties, this changes to 67% for both cards, 29% for 1 card, and 4% for no cards.
In other words, when there is a tie, most of the time the first pair will be in the community cards. For all two pair hands, most of the time the second pair is in the community cards, and with ties, 2/3 of the time the second pair will be in the community cards.
Combining, these results in the table below, we can observe that 83% of time for all two pair hands for Player1, one of the pairs is in the community.
Two Pair: First Pair Cards in Community | Two Pair: Second Pair Cards in Community | Count | Percentage |
1 | 2 | 7166 | 30.78 |
2 | 1 | 6030 | 25.90 |
2 | 2 | 4183 | 17.97 |
1 | 1 | 3859 | 16.58 |
0 | 2 | 1096 | 4.71 |
2 | 0 | 945 | 4.06 |
For two-pair ties, 96% of the time one of the pairs is in the community cards.
Two Pair: First Pair Cards in Community (Ties) | Two Pair: Second Pair Cards in Community (Ties) | Count | Percentage |
2 | 2 | 3003 | 33.91 |
1 | 2 | 2433 | 27.47 |
2 | 1 | 2190 | 24.73 |
0 | 2 | 466 | 5.26 |
2 | 0 | 409 | 4.62 |
1 | 1 | 356 | 4.02 |
We now have an answer to our original question: why is the second pair less important than the kicker in breaking a tie? The answer is that when there is a two-pair kind tie, 96% of the time one of the pairs will be shared. And chances are that it will be the second pair and second pairs tend to be low in rank. And since a low ranked, likely shared second pair is not likely to break the tie, we must move to the kicker in order to do so.
Key Takeaway:
- Don't depend on the second pair to break a tie.
- However, a second pair of 6s or better is needed to win a tie break when depending on the second pair to do so.
- To win a two-pair tie break more than 50% of the time, have at least a first pair of Jacks, at least a second pair of 6s, and a high card of a Queen or better.
- have at least a King high card to break the tie
- One pair: a pair of 9s or better with a King kicker is needed to break the tie in your favor more often than not
- Two Pair: a pair of Jacks, a pair of 6s, and a high card Queen, or better, is needed to break the tie in your favor more often than not
Dealt Hole Card Odds
But how does one know which kind of hand one will have until all the cards have been dealt? One can't know most of the time. So here are some other insights to help in judging whether or not to play or fold.Consider the cards that are dealt to player1 (the hole cards). Below is a heat map of winning percentage against one other player based on the rank of the cards dealt (Note: 14="Ace", 13="King", 12="Queen", and 11="Jack"). I am ignoring suit for simplicity.
Win% | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
2 | 55 | 31 | 29 | 32 | 27 | 34 | 36 | 41 | 44 | 42 | 43 | 47 | 53 |
3 | 32 | 55 | 29 | 31 | 34 | 35 | 34 | 41 | 40 | 44 | 49 | 50 | 56 |
4 | 31 | 29 | 51 | 33 | 35 | 35 | 39 | 36 | 40 | 45 | 47 | 49 | 53 |
5 | 31 | 32 | 35 | 59 | 38 | 38 | 41 | 42 | 44 | 45 | 50 | 54 | 55 |
6 | 33 | 32 | 36 | 38 | 59 | 42 | 40 | 43 | 46 | 47 | 49 | 49 | 60 |
7 | 31 | 35 | 38 | 38 | 40 | 65 | 43 | 48 | 45 | 50 | 49 | 53 | 55 |
8 | 34 | 34 | 38 | 38 | 41 | 46 | 66 | 44 | 50 | 48 | 51 | 56 | 61 |
9 | 39 | 39 | 40 | 40 | 41 | 44 | 44 | 75 | 50 | 52 | 53 | 58 | 61 |
10 | 38 | 39 | 43 | 44 | 44 | 42 | 50 | 51 | 77 | 53 | 57 | 62 | 62 |
11 | 48 | 45 | 44 | 47 | 45 | 47 | 51 | 54 | 51 | 78 | 61 | 61 | 58 |
12 | 44 | 45 | 51 | 47 | 49 | 51 | 53 | 53 | 56 | 57 | 81 | 62 | 62 |
13 | 48 | 51 | 50 | 51 | 54 | 50 | 54 | 58 | 60 | 58 | 56 | 84 | 63 |
14 | 54 | 53 | 56 | 57 | 58 | 58 | 58 | 57 | 61 | 58 | 62 | 63 | 85 |
The heat map changes color from red (low values) to yellow (middle values) to green (high values). The heat map is roughly symmetric along the diagonal, since the order of hold cards doesn't matter (e.g., 14, 13 = 13, 14). The greener, the better.
As we would expect, the higher the cards are, the better the chances of winning are. The exception to this is the diagonal, representing pairs that are dealt as hole cards (e.g., 2,2). These are much better than similar hole card ranks that are not pairs (e.g., 2,3).
For simplicity, let's just look at which hole cards win 50% or more of the time. Here is that map:
Win% | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
2 | 55 | 31 | 29 | 32 | 27 | 34 | 36 | 41 | 44 | 42 | 43 | 47 | 53 |
3 | 32 | 55 | 29 | 31 | 34 | 35 | 34 | 41 | 40 | 44 | 49 | 50 | 56 |
4 | 31 | 29 | 51 | 33 | 35 | 35 | 39 | 36 | 40 | 45 | 47 | 49 | 53 |
5 | 31 | 32 | 35 | 59 | 38 | 38 | 41 | 42 | 44 | 45 | 50 | 54 | 55 |
6 | 33 | 32 | 36 | 38 | 59 | 42 | 40 | 43 | 46 | 47 | 49 | 49 | 60 |
7 | 31 | 35 | 38 | 38 | 40 | 65 | 43 | 48 | 45 | 50 | 49 | 53 | 55 |
8 | 34 | 34 | 38 | 38 | 41 | 46 | 66 | 44 | 50 | 48 | 51 | 56 | 61 |
9 | 39 | 39 | 40 | 40 | 41 | 44 | 44 | 75 | 50 | 52 | 53 | 58 | 61 |
10 | 38 | 39 | 43 | 44 | 44 | 42 | 50 | 51 | 77 | 53 | 57 | 62 | 62 |
11 | 48 | 45 | 44 | 47 | 45 | 47 | 51 | 54 | 51 | 78 | 61 | 61 | 58 |
12 | 44 | 45 | 51 | 47 | 49 | 51 | 53 | 53 | 56 | 57 | 81 | 62 | 62 |
13 | 48 | 51 | 50 | 51 | 54 | 50 | 54 | 58 | 60 | 58 | 56 | 84 | 63 |
14 | 54 | 53 | 56 | 57 | 58 | 58 | 58 | 57 | 61 | 58 | 62 | 63 | 85 |
This more clearly shows which hole card combos win more often (green) and which do not (red). The winning hole cards form an arrow, including the diagonal (all pairs) and the lower right corner. Based on the above, we can offer another set of general principles.
- if your hole cards form a pair (e.g., 2,2), you should play
- if you have an Ace (14), you should play
- for hole cards of rank1 and rank2, if rounding the result of sqrt((rank1)^2 + (rank2)^2) is >= 14, then you should play.
- Note: since this is hard to do mentally, taking the rounded average of rank1 and rank2 >= 10 works almost as well.
Win% | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
2 | 3 | 4 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
3 | 4 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 10 | 11 | 12 | 13 | 14 |
4 | 4 | 5 | 6 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
5 | 5 | 6 | 6 | 7 | 8 | 9 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 11 | 12 | 13 | 13 | 14 | 15 |
7 | 7 | 8 | 8 | 9 | 9 | 10 | 11 | 11 | 12 | 13 | 14 | 15 | 16 |
8 | 8 | 9 | 9 | 9 | 10 | 11 | 11 | 12 | 13 | 14 | 14 | 15 | 16 |
9 | 9 | 9 | 10 | 10 | 11 | 11 | 12 | 13 | 13 | 14 | 15 | 16 | 17 |
10 | 10 | 10 | 11 | 11 | 12 | 12 | 13 | 13 | 14 | 15 | 16 | 16 | 17 |
11 | 11 | 11 | 12 | 12 | 13 | 13 | 14 | 14 | 15 | 16 | 16 | 17 | 18 |
12 | 12 | 12 | 13 | 13 | 13 | 14 | 14 | 15 | 16 | 16 | 17 | 18 | 18 |
13 | 13 | 13 | 14 | 14 | 14 | 15 | 15 | 16 | 16 | 17 | 18 | 18 | 19 |
14 | 14 | 14 | 15 | 15 | 15 | 16 | 16 | 17 | 17 | 18 | 18 | 19 | 20 |
As might be expected, the heatmap of losing percentage looks like the opposite of the heatmap for winning percentage:
Lose% | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
2 | 44 | 63 | 66 | 62 | 66 | 61 | 58 | 55 | 51 | 53 | 52 | 50 | 43 |
3 | 62 | 44 | 67 | 63 | 62 | 60 | 59 | 54 | 56 | 52 | 47 | 45 | 42 |
4 | 64 | 65 | 48 | 59 | 59 | 58 | 55 | 59 | 56 | 50 | 49 | 46 | 43 |
5 | 64 | 61 | 59 | 40 | 56 | 55 | 53 | 54 | 51 | 51 | 45 | 43 | 41 |
6 | 64 | 62 | 57 | 55 | 39 | 53 | 54 | 52 | 50 | 48 | 46 | 48 | 38 |
7 | 63 | 61 | 56 | 56 | 53 | 34 | 53 | 48 | 50 | 46 | 47 | 42 | 41 |
8 | 60 | 59 | 56 | 56 | 53 | 47 | 33 | 51 | 46 | 48 | 47 | 40 | 36 |
9 | 56 | 56 | 56 | 56 | 54 | 51 | 52 | 25 | 47 | 44 | 43 | 39 | 37 |
10 | 56 | 56 | 53 | 52 | 52 | 53 | 45 | 45 | 23 | 43 | 41 | 36 | 35 |
11 | 49 | 49 | 52 | 49 | 51 | 48 | 45 | 42 | 46 | 21 | 38 | 36 | 40 |
12 | 51 | 50 | 45 | 49 | 46 | 45 | 43 | 44 | 41 | 39 | 19 | 35 | 37 |
13 | 48 | 45 | 47 | 43 | 42 | 46 | 41 | 40 | 38 | 39 | 40 | 16 | 35 |
14 | 43 | 43 | 39 | 40 | 39 | 39 | 38 | 41 | 36 | 39 | 37 | 34 | 15 |
Now consider the tying heatmap. One is extremely unlikely to tie with a pair (the diagonal). One is more likely to tie (6-8%) in the lower ranks (2-8) and less likely to tie (1-5%) in the upper ranks (9-Ace).
Tie% | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
2 | 1 | 6 | 5 | 6 | 7 | 5 | 6 | 4 | 5 | 5 | 5 | 3 | 4 |
3 | 6 | 0 | 4 | 6 | 4 | 5 | 7 | 5 | 4 | 4 | 4 | 5 | 3 |
4 | 5 | 6 | 2 | 8 | 6 | 7 | 6 | 5 | 4 | 5 | 4 | 6 | 4 |
5 | 5 | 7 | 6 | 1 | 6 | 7 | 6 | 5 | 5 | 4 | 4 | 4 | 4 |
6 | 3 | 6 | 7 | 7 | 1 | 4 | 7 | 5 | 5 | 5 | 5 | 3 | 3 |
7 | 6 | 4 | 7 | 6 | 7 | 1 | 4 | 5 | 5 | 4 | 4 | 5 | 4 |
8 | 6 | 7 | 6 | 6 | 6 | 7 | 1 | 5 | 4 | 4 | 2 | 4 | 2 |
9 | 6 | 5 | 4 | 4 | 6 | 5 | 5 | 0 | 4 | 3 | 4 | 3 | 2 |
10 | 6 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 0 | 4 | 2 | 2 | 3 |
11 | 3 | 6 | 4 | 5 | 4 | 5 | 3 | 4 | 3 | 0 | 2 | 3 | 2 |
12 | 5 | 5 | 3 | 4 | 5 | 3 | 4 | 3 | 3 | 3 | 1 | 3 | 2 |
13 | 4 | 4 | 3 | 6 | 4 | 5 | 5 | 2 | 3 | 3 | 4 | 1 | 1 |
14 | 4 | 4 | 5 | 4 | 3 | 3 | 4 | 2 | 3 | 3 | 2 | 3 | 0 |
Key Cards
By key cards, I mean the rank of the cards that are used to determine the hand or determine a tie break. For example, a full house of 3-3-3-J-J would have a key card value of 3. If there is a tie with an opponent of 3-3-3-9-9, then the key card value would be 11 (for Jacks). Can looking at hands this way tell us anything useful? Consider the table below:Player1 KeyCards | Lose | Tie | Win | Win% | Lose% | Tie% | Win/(Win+ Lose) | Lose/(Win+Lose) |
14 | 8283 | 493 | 8056 | 47.86 | 49.21 | 2.93 | 49.31 | 50.69 |
13 | 6122 | 387 | 6766 | 50.97 | 46.12 | 2.92 | 52.50 | 47.50 |
12 | 4785 | 412 | 5829 | 52.87 | 43.40 | 3.74 | 54.92 | 45.08 |
11 | 4169 | 437 | 4977 | 51.94 | 43.50 | 4.56 | 54.42 | 45.58 |
10 | 3568 | 460 | 4248 | 51.33 | 43.11 | 5.56 | 54.35 | 45.65 |
9 | 3262 | 480 | 3698 | 49.70 | 43.84 | 6.45 | 53.13 | 46.87 |
8 | 3086 | 519 | 3118 | 46.38 | 45.90 | 7.72 | 50.26 | 49.74 |
7 | 2888 | 435 | 2765 | 45.42 | 47.44 | 7.15 | 48.91 | 51.09 |
6 | 2656 | 282 | 2374 | 44.69 | 50.00 | 5.31 | 47.20 | 52.80 |
5 | 2528 | 137 | 1754 | 39.69 | 57.21 | 3.10 | 40.96 | 59.04 |
4 | 2326 | 70 | 1495 | 38.42 | 59.78 | 1.80 | 39.13 | 60.87 |
3 | 2222 | 44 | 1309 | 36.62 | 62.15 | 1.23 | 37.07 | 62.93 |
2 | 2263 | 66 | 1231 | 34.58 | 63.57 | 1.85 | 35.23 | 64.77 |
The table is sorted by key card rank descending. Here are some observations:
- The higher the rank, the more likely the key card. In other words, key cards are heavily distributed towards the higher ranks, and then decrease towards the lower ranks.
- Excepting the Ace, the Win% decreases by generally following rank, that is, the higher the rank, the higher the Win%.
- A rank of 10 or better is needed to win more than 50% of the time (Aces are the exception).
- The middle ranks tie the most (8 at 8% and 7 at 7%).
What remains to be explained is the Ace. Why is it more likely to lose than win as far as key cards go? The reason is that the Ace is very likely to be the key card for a high card hand coupled with the fact that high card hands lose most of the time. Consider the below table:
Player1 KeyCards: High Card Hand | Count | % of Total |
14 | 7251 | 40.76 |
13 | 4186 | 23.53 |
12 | 2297 | 12.91 |
11 | 1369 | 7.69 |
10 | 801 | 4.50 |
9 | 562 | 3.16 |
8 | 461 | 2.59 |
7 | 391 | 2.20 |
6 | 256 | 1.44 |
5 | 158 | 0.89 |
4 | 59 | 0.33 |
Ace key cards account for 40% of all high card hands. High card hands lose about 82% of the time (from the table near the beginning). So it should be no surprise that a very high percentage of Ace key cards lose. In fact, by observing the table below, we see that this is the only instance in which an Ace key card loses most of the time. Of all the hands in which an Ace is the key card, 78% of the losses occur when the hand is a High Card hand.
Player1 - Ace Key Card | Lose | Tie | Win | Win% |
1.Royal Flush | 0 | 0 | 23 | 100.000 |
5. Flush | 36 | 0 | 880 | 96.070 |
3. Four of a Kind | 0 | 1 | 13 | 92.857 |
4. Full House | 5 | 15 | 193 | 90.610 |
7. Three of a Kind | 43 | 0 | 342 | 88.831 |
6. Straight | 18 | 106 | 480 | 79.470 |
8. Two Pair | 561 | 371 | 2567 | 73.364 |
9. One Pair | 1097 | 0 | 2830 | 72.065 |
10. High Card | 6523 | 0 | 728 | 10.040 |
Key takeaway:
- Key card rank alone is not very useful in determining win% for a hand.
- A key card of rank 10 or better is slightly more likely to win than lose.
Redeeming the One Pair and High Card Hands?
Only the one pair and high card hands lose most of the time when playing against one other player. But are there types of one pair hands and high card hands that win more than 50% of the time? By observing the key cards for these two kinds of hands, we discover that in no circumstance is a High Card hand likely to win more than 50% of the time:Player1KeyCards: High Card Hand | Lose | Tie | Win | Win% |
14 | 6523 | 728 | 10.04 | |
13 | 3584 | 602 | 14.38 | |
12 | 1884 | 413 | 17.98 | |
11 | 1024 | 345 | 25.20 | |
10 | 533 | 268 | 33.46 | |
9 | 331 | 44 | 187 | 33.27 |
8 | 232 | 98 | 131 | 28.42 |
7 | 186 | 114 | 91 | 23.27 |
6 | 127 | 89 | 40 | 15.63 |
5 | 93 | 49 | 16 | 10.13 |
4 | 35 | 24 | 0.00 |
The best is 33%. One is still better off folding in any circumstance if one only has a high card hand.
What about the one-pair hand? Turns out, a one pair hand of Kings or better wins more than 50% of the time:
Player1KeyCards: One Pair Hand | Lose | Tie | Win | Win% |
14 | 1097 | 2830 | 72.07 | |
13 | 1773 | 2407 | 57.58 | |
12 | 2072 | 89 | 2160 | 49.99 |
11 | 2254 | 176 | 1789 | 42.40 |
10 | 2154 | 254 | 1546 | 39.10 |
9 | 2087 | 268 | 1359 | 36.59 |
8 | 2019 | 253 | 1170 | 33.99 |
7 | 1958 | 198 | 1047 | 32.69 |
6 | 1883 | 94 | 875 | 30.68 |
5 | 1886 | 43 | 807 | 29.50 |
4 | 1805 | 12 | 745 | 29.08 |
3 | 1812 | 702 | 27.92 | |
2 | 1889 | 726 | 27.76 |
So one shouldn't always fold if one has a one pair hand. Instead, one should play with an Ace or King pair hand, and probably also with a Queen pair hand (wins more than loses).
We can update the win, tie, loss by hand table to further refine our guidance. It is reflected below.
Conclusion
Here is an updated table with win ratios by type of hand:Player1 | Lose | Tie | Win | Lose/All | Tie/All | Win/All |
1.Royal Flush | 0 | 0 | 23 | 0.00 | 0.00 | 1.00 |
2. Straight Flush | 0 | 4 | 141 | 0.00 | 0.03 | 0.97 |
3. Four of a Kind | 11 | 5 | 158 | 0.06 | 0.03 | 0.91 |
4. Full House | 160 | 182 | 2300 | 0.06 | 0.07 | 0.87 |
5. Flush | 352 | 89 | 2380 | 0.12 | 0.03 | 0.84 |
6. Straight | 150 | 768 | 3050 | 0.04 | 0.19 | 0.77 |
7. Three of a Kind | 1165 | 157 | 3596 | 0.24 | 0.03 | 0.73 |
8a. Two Pair - 5s or better | 6101 | 1187 | 14495 | 0.28 | 0.05 | 0.67 |
9a. One Pair - Queens (12) or better | 4942 | 89 | 7397 | 0.40 | 0.01 | 0.60 |
9b. One Pair - Jacks (11) or worse | 19747 | 1298 | 10766 | 0.62 | 0.04 | 0.34 |
8b. Two Pair - 4s or worse | 978 | 25 | 493 | 0.65 | 0.02 | 0.33 |
10. High Card | 14552 | 418 | 2821 | 0.82 | 0.02 | 0.16 |
Here is summary of the suggestions for playing against one other player:
- If one has a two pair 5s or better, one pair Queens or better, or three of a kind or better, then play
- If one has a two pair 4s or worse, one pair jacks or worse, or a high card hand, then fold
- Ties:
- Depend only on a King or better high card to break the tie in your favor
- One pair: 9s or better to win the tie break most of the time
- Two pair: Jacks or better to win the tie break most of the time
- Hole cards
- If your hole cards form a pair (e.g., 2,2), you should play
- If you have an Ace (14), you should play
- For hole cards of rank1 and rank2, if rounding the result of sqrt((rank1)^2 + (rank2)^2) is >= 14, then you should play.
- Note: since this is hard to do mentally, taking the rounded average of rank1 and rank2 >= 10 works almost as well.
Good luck!
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