Poker Trouble Spots: How to Play Second Pair

Glen Chorny

Even though it would be nice to flop the nuts in every hand you play, in reality you're going to come up short more often than not.

And 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.

Regardless of how you play the game, poker is based in mathematics. For that reason alone, the math behind second pair is the first place to start.

The Odds of Second Pair

To keep things simple, it's best to start with a very cut-and-dried example. You're in middle position with one player in front of you, one player behind you.

Flop:      

Your Hand:    

In this scenario there are a variety of numbers you need 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. 

William Hung
Sometimes you just have to bang it out.

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 gives you a solid idea of exactly how strong your hand really is.

But what about 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 preflop. Because we can see five cards (our two cards, plus the five on the board), we know that only three kings where available to be dealt out of 47 cards.

This might seem tricky, because preflop there were 52 cards to deal from. But now we know that no player was dealt any of the cards we see. Thus, we can be 100% certain that no player was dealt them, and we can take them out of our equation.

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 preflop, 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

birthday paradox
The Birthday Paradox.

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

This may be shocking to people who are not well versed in probability - and that sums up the vast majority of us.

We find this phenomenon frequently in the realm of probability theory: it's very similar to the famous birthday problem, or birthday paradox.

Simply put, the birthday problem proves that if you have 23 randomly chosen people in one room, there is 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 to you, 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 nonmatches.

Your odds getter better on each try, and even though the individual odds are small, the odds from every attempt accumulate, giving you the above result.

If you'd like to learn more about the birthday paradox, including the really neat math behind charting the results, you can find it on Wikipedia here.

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:

The Scenario:

Flop:      

Your Hand:    

We calculated a 77% chance of another player having been dealt a king preflop. If every player plays every hand they're dealt to the flop, there is 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. 

Ivan Demidov
Warning: Maths may cause cranial discomfort.

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 preflop, 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 preflop:

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, saying 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

Phil Ivey
The best math is always getting to count the pot you just won.

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 holding hands such as these.

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 is 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, meaning 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, 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 and prone to do so, you should be willing to fold away 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.

More Beginner Poker Strategy Articles:

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James 2012-08-18 16:55:13

I stopped reading this article almost immediately, the scenario doesn't even describe how the hero got to the flop, just that we are between two players....rest of the article is useless without knowing the pre-flop action.

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