Poker Trouble Spots: Second Pair Part 1

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 Numbers

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.


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.

William Hung
Sometimes you just have to bang it out.

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 a dog 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%.

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.

In part two of this article, we'll look at how to interpret these numbers, what they really mean to your hand and get some idea of how to actually play these hands.

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