Even though it would be nice to flop the nuts in every hand, in reality you'll come up short more often than not.
Because the vast majority of hands you'll play won't be the nuts, or obvious best hands, you have to play well enough to maximize the value of high-marginal hands.
A hand such as second pair holds a lot of value but can be very difficult to play - especially if you're sitting in middle position.
It makes no sense to immediately throw away as valuable a hand as this. But you don't want to get caught up committing your stack into a large pot with it either.
Odds for Second Pair
Regardless of how you think of the game, poker is really based in mathematics. For that reason alone the math behind second pair is the first place to start.
To keep things simple it's best to start with a very cut-and-dried example:
Flop: K K 10 10 5 5
Your Hand: 10 10 A A
In this scenario there are a variety of numbers to take into consideration before you can evaluate where you stand; the first is simple equity. If there are nine other random hands with you on this flop, and all hands go to the river, you're 17% to win the hand.
This makes you an underdog against the field but almost twice as likely to win as any other single player.
Although an equity example like this isn't realistic it does give you an idea of how strong your hand really is.
What About a King?
But what if another player has a king? If another player is holding ace-king here (still with eight other random hands), you're now only 6% to win the hand.
Even if the other player holds king-deuce your equity drops to only 13%. This leads us to our next question:
- What are the odds another player was dealt a king?
Since we're asking this question on the flop we know that only one to three kings could have been dealt to players pre-flop. Because we can see five cards (our two cards plus the five on the board) we know only three kings were available to be dealt out of 47 cards.
This might seem tricky because pre-flop there were 52 cards to deal from. But now we know that no player was dealt any of the cards we see. So we can be 100% certain that no player was dealt them and we can take them out of our equation.
How Do We Know?
Even though at the time of the deal the cards on the flop were just as likely to be dealt to a player as any other cards we can clearly see that they weren't.
The way to find out the probability of at least one player being dealt a king is by using the following equation:
- (44/47) * (43/46) * (42/45) * (41/44) * (40/43) * (39/42) * (38/41) * (37/40) * (36/39) * (35/38) * (34/37) * (33/36) * (32/35) * (31/34) * (30/33) * (29/32) * (28/31) * (27/30) = %
This equation represents the fact that when the first card was dealt to a player (who wasn't you) there were 44 cards in the deck that were not a king or were not going to be used on the flop or in your hand. Assuming the first card dealt was not a king the next card dealt has only 43 cards out of 46, and so on.
By multiplying the odds of each card dealt together for all 18 cards dealt to other players pre-flop, we reach a final percentage of the chance that a king was not dealt.
- 0.936 * 0.934 * 0.933 * 0.932 * 0.930 * 0.929 * 0.927 * 0.925 * 0.923 * 0.921 * 0.919 * 0.917 * 0.914 * 0.912 * 0.909 * 0.906 * 0.903 * 0.9 = 0.225
The odds that a king was not dealt = 23%. Since 100%-23% = 77%, we now know that the odds that a player was dealt a king are 77%.
The Birthday Paradox
The odds above are similar to the birthday paradox, which is usually shocking to people not well versed in probability - and that's the vast majority of us.
Simply put the birthday paradox says that if you have 23 randomly chosen people in one room, there's a 50% chance that two of them will have the same birthday.
Make it 60 people in that room and the probability rises to a staggering 99%. If this seems unreasonable you simply have to consider that every time you add a new birthday to the list you have a larger pool of possible matches against a smaller pool of possible non-matches.
Your odds getter better on each try, and even though the individual odds are small the odds from every attempt accumulate to give you the above result. If you' want to learn more about the birthday paradox, including the math behind charting the results, you can find it on Wikipedia.
How to Play Second Pair on the Flop
Now we can put those numbers into play to get some general guidelines for how to act (and react) with second pair on the flop. The idea of pot control and reserving big pots for big hands should be ingrained in your mind. Here's a quick read on the subject:
Flop: K K 10 10 5 5
Your Hand: 10 10 A A
We calculated a 77% chance of another player having been dealt a king pre-flop. If every player plays every hand they're dealt to the flop, there's only a 23% chance that you have the best pair. The chances of you having the best hand are even lower after allowing for trips and two-pair scenarios.
The first thing to understand is that the 23% chance of another player having a king does not translate into you having a 23% chance at winning the pot. Your equity in the pot is only 17%.
"How likely is it that another player has a king on the flop?"
This is the most important question of all. We know that there's a 77% chance of another player having been dealt a king pre-flop, but what are the chances that a player has called the bets to take their king to the flop?
Although every player is different, and a player's opening range will change depending on many factors, we can make a general chart of all the hands with a king grouped by whether or not they would have been played pre-flop:
|Folded||Maybe Played||Definitely Played|
|K-2 off to K-9 off (96)||K-2 suited to K-9 suited (32), K-T off (12)||K-T suited (4), K-J - K-A all (64)|
Number in parentheses = the total number of permutations in that range.
Total Folded: 96
Total Potentially Played: 44
Total Definitely Played: 68
This very basic chart is not an accurate look at how every specific player feels about all of these hands but more of a generalization as to how the hands are viewed as a whole, by pros and fish alike.
Luckily, we don't need accurate numbers for this example; approximations will do us just fine. For the sake of making things easy we'll chop the maybes right down the middle and say half of them would get played while the other half would be folded.
- Total Hands with a King = 208
- Total Hands Played = 90
- 90/208 = 43%
Second Pair is Strong on Dry Board
If 77% of the time a player was dealt a king, and out of those hands 43% of the time the king was played to the flop, the chances of a player having a king on the flop are somewhere around 33%.
This means, aside from a random two pair or trips, you have the best hand on the flop close to 67% of the time.
Naturally, this number will change dramatically depending on the style of players and game (if the game is very loose your chances go down and vice versa), but in general this is a very solid place to start from.
Poker professionals understand how powerful a strong second pair is on a dry board. This is why they can be seen making large bets and calls on TV with hands like second pair.
If you're against a opponent you know to be very tight (meaning they will only play a small number of possible king hands to the flop), you can almost count on having the best hand on the flop 67% of the time.
Fold Second Pair to Signs of Strength
Even though you have the best hand more often than not, there's simply no other possible hand your opponent could have that you beat and that they would want to make or call bets with.
A second-pair hand should almost always be played for a quick win of a small pot.
Any players willing to invest into a large pot against you simply have to have a better hand or a very strong draw. On a board as dry as in our example, it doesn't make any sense for them to have a draw so any player willing to play back at you either has you beat or is bluffing.
Although players do bluff, bluffing is far less common than many beginning poker players seem to think and especially at the low-limit games.
Unless you have a read on the player and know that they're capable of making bluffs against you, you should be willing to fold your second pair at the sign of significant strength from your opponent.
On a dry board you typically will have the best hand in play with a high-kicked second pair. You should feel confident using these hands to take down small pots. If however you get called after betting the flop, you generally want to shut down and give up unless you improve on the turn.