What follows isn't anything you should memorize. It's more of a study into some common situations you might be curious about. Maybe the numbers can help you develop you strategy a little bit.
Photo courtesy of Brian Lary
Kings running into aces
"Every time I pick up a pair of kings, some moron has aces! Online poker is so rigged."
Heard it before? Seems to happen all the time? As a matter of fact, how often does it happen, really? Let's have a look at the numbers.
When you hold a pair of kings, the chances that a specific opponent has a pair of aces are about 1/204. That particular player will have a pair of aces once every 204 times you have a pair of kings.
However, most of the time there's more than one player in the hand. At a full table there are nine of them. What's the probability that one of them has a pair of aces when you have kings?
With nine players, this probability is around 1/15. So, there you go. When you pick up kings, you'll run into aces one time in fifteen, give or take.
That's hardly "all the time", now is it? As a matter of fact, if you can pull out a few thousand hand histories from a poker site and show that kings meet aces clearly more often than this, you could take your "online poker is rigged" to court and win a historical victory against the online poker industry.
Then again, if you put in a pre-flop raise with kings and an opponent with a huge stack insta-shoves behind you, chances that he has aces go up considerably.
His big re-raise is a new piece of information and it should affect your analysis. Read more in our article on how to update hypotheses.
The final decision is always up to you, I'm glad to say.
Note: The probability that your aces run into pocket kings is exactly the same. It goes both ways. Think of that when you cry 'rigged'.
Flopping a set
You take a flop with a pair of sixes, hoping to flop a set. What are the chances?
Two cards are in your hand. Of the remaining 50 cards, only two will give you a set (the remaining sixes). We have the following calculation:
P(set) = 1 - P(no set) = 1 - 48/50 * 47/49 * 46/48 = 0.12
In other words, you'll flop a set once in eight trials, give or take. So to make this play profitable, you probably should be able to win eight times the size of the pre flop bet. Which is not obvious even if you're playing with deep stacks. Oftentimes, the opponent won't pay off your set.
Of course, you may be able to pick up a few pots with your unimproved pair as well. And sometimes your pair will be the best hand right from the start. Then on the other hand, you won't win every time you flop a set.
The point is, you need a certain leverage to call pre-flop raises with small pairs and be a winner in the long run. You probably want to save the play for specific situations where it actually makes sense.
By the way, if you have a non-paired starting hand such as 76, the probability of flopping two pair by pairing both your hole cards is only 2.1%. On the other hand, you get this kind of hand twelve times as often as a paired hand, so flopping two pair is more common than flopping a set (even though you probably fold a larger percentage of your non-pairs compared to pairs).
Pocket pair running into a set
On the other hand, when you raise it up pre-flop with a high pair and get a caller, you will fear opponents hitting sets with their lower pairs.
But as we saw above, you shouldn't go all paranoid about a lower pair hitting a set against you. Seven times out of eight, he won't.
With several callers, the risk goes up, though. Let's say all callers have a pair and no one has the same pair (two simplifications).
The chances that at least one of them hits a set against you go like this:
Opponents | Risk of set (%) |
1 | 12 |
2 | 23 |
3 | 32 |
4 | 40 |
5 | 47 |
6 | 54 |
7 | 59 |
8 | 64 |
9 | 68 |
Even with as much as 5 opponents on the flop, you're still a favorite not to be up against a set.
Then on the other hand, we haven't said anything about an opponent holding a higher pair to start with. See next section.
Pair meeting higher pair
A pair is a nice starting hand, but if an opponent has a higher pocket pair you're in trouble. What are the chances for that? Let's assume you're ten players at the table, to be on the safe side.
The risk that an opponent holds a pair that's higher than yours go ilke this:
| Risk of |
AA | 0 |
KK | 4 |
8 | |
JJ | 12 |
TT | 16 |
99 | 20 |
88 | 23.5 |
77 | 27 |
66 | 30 |
55 | 33 |
44 | 36 |
33 | 39 |
22 | 42 |
As you can see, these risks are not gigantic. And if there are fewer opponents, the chances go down even more. Most of the time, a pair is the best hand pre-flop.
Basically you could move in with any pair and have positive EV. Then, of course, IF you get a call, high pairs are much more likely than a random hand. Also, remember that two overcards have good equity against your lower pair (around 50%). With that strategy, you'd probably be winning small pots and losing big ones.
Overcards to your pocket pair
You take a flop with a pair in your hand and hope not to see an overcard on the flop. What's the risk of overcards on the flop?
Well, it depends on the strength of your pair, as follows:
| Risk of |
AA | 0 |
KK | 23 |
41 | |
JJ | 57 |
TT | 69 |
99 | 79 |
88 | 87 |
77 | 92 |
66 | 96 |
55 | 98 |
44 | 99 |
33 | 99.9 |
22 | 100 |
Note how quickly the risk of overcards grows. With a pair of jacks, you're already an underdog to be holding an overpair to the flop. That's probably why a pair of jacks is so hard to play.
Of course, the risk that someone actually does flop a higher pair is something else. It's much lower than the numbers above, since they need to have a matching card in their hand.
But the fact that the board has an overcard will put a damper on your game, and if an opponent heats up the action, you'll be looking at some hard decisions.
The chances of flopping a set with your pair are around 12%, so you're always more likely to run into overcards than flopping a set. Except with pocket rockets, of course.
Think of that before you go into the set-mining business.
/Charlie River
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