Why does a flush beat a straight?

[Wacken]

Oh, i should have added a sarcasm sign to my post.

[memphis57]

Just wait until we get around to wild card games - in deuces wild 5 card draw, you get many more 4-of-a-kinds than straights or flushes, and yet they beat them.

[blank frank]

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Would it change anything if you were to enumerate the potential 5 card holdings from 7 cards? My guess is that it would. A straight seems as though it would become relatively more likely (compared to a straight) as more cards are added.

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Actually, while the probability ranking of hands changes with seven cards, the ranking of straight and flush remain the same. What happens is that high card becomes rarer than either pair or two pair. Flush and straight get much closer (3.0% to 4.6%), but straights are still more common. In fact, except for high card, the rankings stay the same up to ten cards dealt. Of course, at that point high card is rarer than four of a kind. On the other hand, this gets a little fuzzy. With seven cards you could have a straight and a flush, and yet not have a straight flush, so if you are judging the relative probabilities, how do you call it? This gets worse the more cards you have. My calculations assume the rankings, so may be biased towards them.

[jba]

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Would it change anything if you were to enumerate the potential 5 card holdings from 7 cards? My guess is that it would. A straight seems as though it would become relatively more likely (compared to a straight) as more cards are added.

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Actually, while the probability ranking of hands changes with seven cards, the ranking of straight and flush remain the same. What happens is that high card becomes rarer than either pair or two pair. Flush and straight get much closer (3.0% to 4.6%), but straights are still more common. In fact, except for high card, the rankings stay the same up to ten cards dealt. Of course, at that point high card is rarer than four of a kind. On the other hand, this gets a little fuzzy. With seven cards you could have a straight and a flush, and yet not have a straight flush, so if you are judging the relative probabilities, how do you call it? This gets worse the more cards you have. My calculations assume the rankings, so may be biased towards them.

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good points


another note is that with only four cards, a four card straight is more likely than a 4 card flush. with three cards a straight is much more likely than a flush.

[phish]

cause it looks prettier

[pudley4]

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Because once you make your two pair, straight, or flush, it comes down to a matter of ranking your cards, not necessariliy the probability of specific hands, because per your above example there are as few possibilities of making AA22x as there is of making 3344x.

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No. I just threw out AA22x as an example, my point is it is easier to get aces up than it is to get fours up.

A better example may be to show you

KKQQx

which is beat by

AAKKx
AAQQx
AAJJx
....
AA22x


as you can see, there are many more combinations of aces up than there are of kings up. yet, aces up is a better hand.

KKQQx *should* beat AA22x, but it doesn't.

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No, it "shouldn't". The rankings of the different hands (one pair, two pair, straight, etc) are based on their frequency - the lower the frequency, the higher the hand rank.

Within these hands though, there must be some way of determining which hand is "best" (best two-pair hand, best straight, etc). Every two pair hand has the exact same probability (you'll see exactly as many AAKK hands as you will 6644 hands, or AA44, or KK66 hands). So in order to rank them, we have to come up with an arbitrary rating system. As soon as you decide on which pair is higher, you automatically make it the most common two-pair hand (because by definition, everyone is trying to make the best hand, so they'll always use their highest pair as the defining pair in their hand).

If you say "You see more Aces-up hands than Kings-up hands, so Kings-up should win", then next time I get AAKK and you get KKQQ, and you say you should win, I'm going to change my hand to KKAA - I now have Kings-up, and my AA beats your QQ. Or, when I have AA22, and you have KKQQ, your Kings-up is "less likely", so you win; now I'll change my hand to 22AA, and since twos-up is "less likely", I now win.

This circular reasoning is why "intra-group" hand rankings aren't defined the same as "inter-group" hand rankings.

[jba]

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KKQQx *should* beat AA22x, but it doesn't.

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No, it "shouldn't".


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yes, it *should* [img]/images/graemlins/smile.gif[/img]

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...
Within these hands though, there must be some way of determining which hand is "best" (best two-pair hand, best straight, etc). Every two pair hand has the exact same probability (you'll see exactly as many AAKK hands as you will 6644 hands, or AA44, or KK66 hands). So in order to rank them, we have to come up with an arbitrary rating system. As soon as you decide on which pair is higher, you automatically make it the most common two-pair hand (because by definition, everyone is trying to make the best hand, so they'll always use their highest pair as the defining pair in their hand).

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agreed so far

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If you say "You see more Aces-up hands than Kings-up hands, so Kings-up should win", then next time I get AAKK and you get KKQQ, and you say you should win...

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I'll stop it there. In my world, AAKK would still beat KKQQ. I said AA22 loses to KKQQ; AAQQ would beat KKQQ also, but AAJJ would lose to KKQQ.

The *correct* way should be that within two pair hands, the highest low pair *should* win. with AA22 vs KKQQ, the highest low pair is QQ, so KKQQ wins. AAKK vs KKQQ becomes KK vs QQ, so AAKK wins. This would be the statistically correct way to play.

likewise with flushes: A6543 should lose to KQJT8, because the highest low cards are 3 vs 8, 8 wins.

this is why in low games 23457 beats A2348 -- the lowest high card wins.


edit to add: in other words, the "correct" way to look at 2 pair would be to say kings down beats queens down. kings down is the best two pair hand, as there is only one way to get it (AAKKx). queens down is second best, as there are two ways (AAQQx KKQQx), etc. know what I mean?

[danzasmack]

If my memory serves me, a straight originally beat a flush a long time ago - then math was done and error was fixed.

real reason - flush is prettier.

[JKDStudent]

Selective memory tells us that straights are less common than flushes in Hold 'Em, which is probably where this question originally stemmed from. Why is that, if there are twice as many straights than flushes (in a five card game, anyway)? Because you have to base your hand on two cards. Sure, 53o can make 12 straights, but who really plays it? A lot of those straight possibilities are taken out.

So I looked at my Poker Tracker stats. It's only 20k hands, but still. If I look at all hands, including those I folded, 644 of those hands made a straight with the board, and 422 made a flush (keep in mind, this will be off considering the many hands that were over before the river, but still a pretty good representation of relative occurances).

Interestingly enough, if I look at hands that I didn't fold, there were still more straights. 141 to 112. What I didn't expect, both my straights and flushes win 80% of the time. Low sample size, but still interesting to me. I guess it's surprising because it seems like everytime I flop a straight, I lose that hand if it goes to showdown. That pesky selective memory again.