Interesting "paradox" concerning suited cards

[MNpoker]

Well there is definetely some double counting.

Any board of QJT and and no pair wins in either situation.
(i.e. If there is a flush or not isn't relevant)

[WhiteWolf]

[ QUOTE ]
Well there is definetely some double counting.

Any board of QJT and and no pair wins in either situation.
(i.e. If there is a flush or not isn't relevant)

[/ QUOTE ]
And probably more commonly, almost all boards with an Ace or King and no Ten wins in either case, flush or no flush. I'm not sure how to figure the odds on this, or if it explains most of the "discrepancy".

It's not that a flush is going to be beaten so many times: it's that there are many boards that AKs will make a flush with, and although AKo will not make a flush, it will still beat TT some other way.

[BruceZ]

[ QUOTE ]
[ QUOTE ]
Well there is definetely some double counting.

Any board of QJT and and no pair wins in either situation.
(i.e. If there is a flush or not isn't relevant)

[/ QUOTE ]
And probably more commonly, almost all boards with an Ace or King and no Ten wins in either case, flush or no flush. I'm not sure how to figure the odds on this, or if it explains most of the "discrepancy".

It's not that a flush is going to be beaten so many times: it's that there are many boards that AKs will make a flush with, and although AKo will not make a flush, it will still beat TT some other way.

[/ QUOTE ]

Correct, MNPoker and WhiteWolf got it. Very good.

The AKs does indeed make a flush on 5% more hands than AKo, and it does win essentially all of those. But that doesn't mean that those extra winning flushes that make up 5% of the hands will add 5% to the percentage of winning hands, because some of the boards that make AKs a flush would have been won by AK even if it were unsuited, perhaps by only a pair.

As the problem stated, the times that the flush loses to a full house or quads is much too small to account for the issue here. The tens make a full house about 0.8% of the time total, but the times that it makes a full house AND AKs makes one of these extra flushes is on the order of 0.8% of 5% or 0.04%, which is much too small to account for the difference of 1.6% that we were trying to explain.

This isn't a true paradox of course, only a fallacy. It is a paradox in the same way as Simpson's paradox, which is really just a counterintuitive arithmetic fact until you think about it right. It's really an illusion of the mind. Until someone points out the answer, it seems that 5% extra winning flushes has to mean 5% extra wins. It's only when we consider the boards that make up that 5% that we can see the trick, and our minds don't necessarily go there right away.