No, they're not. For example, AK is just as likely as 76, but both those hands are almost three times as likely as AA or 33. How come?

It's got to do with combinatorics; the number of ways that a hand can be constructed using the cards in the deck.

Basic combinatorics of poker hands

For a pair, like AA, there are four cards in the deck that can be used - the four aces. So, the first card of the pair can be picked in four ways. Then there are three aces left, so the second card can be picked in three ways.

In total this gives us 4*3=12 ways to pick two aces from the deck. BUT the order of the cards doesn't matter. Ah-Ad is the same hand as Ad-Ah. To find the true number of AA combinations, we must throw away all duplicates.

To sum up, there are six different AA. If you're not convinced, we can list them:
As-Ad
As-Ac
As-Ah
Ad-Ac
Ad-Ah
Ah-Ac

This is the complete list of AA combinations in a 52 card deck. If you can find another one, you'll get insanely rich.

Of course, if we look at another pair, like TT och 55, it's the same thing. All pairs are made equal. There are six combinations of each pair.

On the other hand, two cards of different face value come in more combinations. Take AK as an example. There are four aces to chose from, and once you've chosen an ace there are four kings to pick among. This gives us a total of 16 AK hands.

Just to be safe, let's list them too:
As-Ks
As-Kh
As-Kd
As-Kc
Ah-Ks
Ah-Kh
Ah-Kd
Ah-Kc
Ad-Ks
Ad-Kh
Ad-Kd
Ad-Kc
Ac-Ks
Ac-Kh
Ac-Kd
Ac-Kc

As you can see, four of these combinations are suited. Twelve are unsuited.

Again, this number is valid for any two cards of different face vale, like KT, A8 or 72.

The important lesson here is that as soon as the cards have been dealt, some hands are more likely than others. If you think for some reason that an opponent has either AA, KK, QQ or AK, you must realize that AK alone is almost as likely as the rest of the hands together - 16 vs. 18 combinations.

Known cards change the combinatorics

So far we've looked at the situation before any action has been taken and before any cards are known. For each card you get to know, the likelihood that the opponent holds a certain hand will change.

For example, if you have an ace in your hand, the number of available AA combinations for the opponent changes. Instead of four aces to pick from, now there are only three.

We pick the first ace among those three, and then there are only two aces left. We pick the second ace from those two to make a pair of aces. So, the number of combinations is 6, but again we must disregard all duplicates since the order of the cards doesn't matter.

With one ace gone (known), there are only three combinations of AA left among the unknown cards. If you hold Ah, for example, only the following three pairs of aces are possiböe for your opponent:
As-Ad
As-Ac
Ad-Ac

With an ace in your hand, the number of AK combinations for the opponent's hand also changes. Three aces and four kings give us 12 combinations. Now AK is four times as likely as AA in your opponent's hand.

When more cards are displayed

If you have an ace in your hand and another ace comes on the flop, the numbers change again. Now there are only two cards left in the deck, so there's only one possible combination of AA.

For AK there are two aces and four kings which gives us 8 AK combinations. The relation between AA and AK is now 1 to 8.

Now, if there are no jacks either in your hand or on the board, there are still six JJ combinations for your opponent, so he or she is is six times as likely to hold JJ as AA, and almost as likely to hold JJ as AK.

And so it goes on. The basic frequencies of hole-card combinations change with each card that is displayed. You need to take this into consideration when you put opponents on a range and try to figure out your winning chances against that range.

/Charlie River

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