More On The Silly Subject Of Hand Rankings

[George Rice]

Many many years ago I sat down with pen, paper and calculator and confirmed the hand rankings as just for five cards, six cards and seven cards. But I did notice that as the number of cards increased, certain hands increased in likelihood at a faster rate than others.

One of the posters sugested that flushes should be ranked lower than straights in holdem. Of course this is false. However, in card games with more cards this would be true. I don't remember if this happens with eight, nine or ten cards, but it does happen. Probably not eight or it might effect the rankings in draw poker (Although having eight cards to choose from is different than discarding three and drawing.). In Omaha, if you were allowed to use all four cards in your hand, then straights possibly should be ranked higher than flushes. Pineapple would be another game where this is close.

Another hand that overtakes its next higher hand is the no pair hand. I don't remember how many cards it takes to happen. It's probably in the same area.

Without doing any fancy probability calculations you can consider the following:

If I deal you 14 cards you are guaranteed to be holding a pair (ignoring any other hands you might hold).

If I deal you 15 cards you are guranteed to be holding either two pair or three of a kind.

If I deal you 17 cards you are guaranteed to be holding a flush.

It would take the same 17 cards to be guranteed to show me two different pairs.

The number of cards where you are guranteed to be holding three of a kind is the same number of cards you need to be guranteed to be holding a full house. That number is 27.

It takes 40 cards to be guranteed that you will be dealt four of a kind.

It takes a whopping 45 cards to be guranteed that you will be able to show me a straight. It takes the same 45 cards to be guranteed a straight flush too.

It takes 49 cards to be guranteed the royal flush.

Of course none of this is applicable to cards as played in casinos today. But you can see why straights and no pair hands become more rare than other hands as the number of cards increases.

Understanding this concept might help in certain home games where there are replacements and/or many board cards.

[BruceZ]

One of the posters sugested that flushes should be ranked lower than straights in holdem. Of course this is false.

That wasn't based on the frequency of straights and flushes in 7 cards, but on the relative difficulty of making them in Hold 'Em. True, if you take all starting hands to the river, you will make more straights than flushes, but that strategy gets expensive. There are more starting hands that make straights than make flushes, and that means there are more starting hands that make straights that you don't play because the chance of hitting a hand is so low. Therefore many of the possible straights that you could have made can't get made, and it becomes more rare to make a straight than a flush. So should you be rewarded when you actually make one?

[Copernicus]

It is more rare to make a kangaroo straight than it is a straight. should that be rewarded?

[George Rice]

That wasn't based on the frequency of straights and flushes in 7 cards, but on the relative difficulty of making them in Hold 'Em.

No, but it should have been.


So should you be rewarded when you actually make one?

Yes, you should win the pot, unless you are beaten by a better hand ;-)


You being a math dude and all I figured you of all posters would appreciate that straights *should* beat flushes if more cards were available, like in some home games.

But try convincing your buddies of that!

[daryn]

it is even more rare to make a kangaroo get into your van willingly

[Wake up CALL]

[ QUOTE ]
it is even more rare to make a kangaroo get into your van willingly

[/ QUOTE ]

Please don't call me willingly. /images/graemlins/smile.gif

[thylacine]

Someone said that with 7 cards, pair is more common than no-pair, so no-pair should be ranked higher, since it is less likely.

BUT WAIT!! When deciding what hand you've got you must choose your best five card hand . But who's to say what's best. Well of course a pair is better than no-pair, isn't it? So if you have a pair in your seven cards you have to use it, right? And that's more likely than no pair at all.

So no-pair is better than pair.

But hold on a minute there. We have to choose our best five card hand from seven, and we now know that no-pair is better than pair. Now of course it's very likely that some five card hand from seven is a no-pair hand, which beats a crummy pair hand as we decided.

But if no-pair is more likely than pair, then by standard poker convention pair is better than no-pair.


BUT WAIT!! When deciding what hand you've got you must choose your best five card hand . But who's to say what's best. Well of course a pair is better than no-pair, isn't it? So if you have a pair in your seven cards you have to use it, right? And that's more likely than no pair at all.

So no-pair is better than pair.

But hold on a minute there. We have to choose our best five card hand from seven, and we now know that no-pair is better than pair. Now of course it's very likely that some five card hand from seven is a no-pair hand, which beats a crummy pair hand as we decided.

But if no-pair is more likely than pair, then by standard poker convention pair is better than no-pair.


BUT WAIT!! When deciding what hand you've got you must choose your best five card hand . But who's to say what's best. Well of course a pair is better than no-pair, isn't it? So if you have a pair in your seven cards you have to use it, right? And that's more likely than no pair at all.

So no-pair is better than pair.

But hold on a minute there. We have to choose our best five card hand from seven, and we now know that no-pair is better than pair. Now of course it's very likely that some five card hand from seven is a no-pair hand, which beats a crummy pair hand as we decided.

But if no-pair is more likely than pair, then by standard poker convention pair is better than no-pair.

etc. etc. etc.

[George Rice]

Good point. I never figured the liklihood to ge guaranteed a no pair hand. Nor do I intend to. /images/graemlins/tongue.gif

Awe, what the heck. The number of cards dealt that gurantees that you will be dealt a no-pair hand (straight and flushes work against you, such as in deuce-seven lowball) is:

Fourteen. So when a pair is guranteed, so is a no-pair hand.

Does the paradox you point out apply to the straight/flush situation? I don't think so. A no pair hand is simply the lack of any other defined hand, and will usually be present with many cards even when you hold another hand (can anyone say high/low poker?).

But I shouldn't have been so quick to assume that a no pair hand will overtake a pair, especially since I didn't actually figure do the math. /images/graemlins/grin.gif However, the conclusion that flushes would overtake straights in liklihood wasn't based on the observations listed at the bottom of my post, although it may have caused me to look into it. It was based on the liklihood of those hands occuring in a given number of cards.

[thylacine]

I didn't crunch any numbers either, but I pretty sure this kind of thing would happen quite generally.

The following will happen `a lot'.

Suppose hands of type A and hands of type B each consist of k cards. Now suppose you get `enough' (enough to get the conclusion) say n cards, and you have to pick the best k card hand.

If n is big enough and it is not messed up by the other possible types of hands then:

If hands of type A are deemed to be better than hands of type B, then it will be more likely that the best hand is a hand of type A rather than a hand of type B.

If hands of type B are deemed to be better than hands of type A, then it will be more likely that the best hand is a hand of type B rather than a hand of type A.

In other words the higher ranked hand is easier to get than the lower ranked hand, no matter what the original ranking was. This is because if n is big enough, then there are all kinds of k card hands, but the crucial point is that you have to choose the `best' k card hand.