Interesting "paradox" concerning suited cards

[BruceZ]

Someone asked me a question today that really had me stumped for awhile until I realized what was going on was very simple. I wonder how many of you will be similarly stumped. Pzhon, Aaron, Siegmund and their ilk, let others have a chance to answer.


Problem

A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/club.gif[/img] beats T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] about 42.6% of the time. A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/spade.gif[/img] beats T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] about 46.0% of the time. So being suited increases AK's chance of winning by 3.4%. The advantage of being suited is that the AKs makes a flush by the river about 7.2% of the time, compared to AKo which makes a flush about 2.2% of the time. AKs makes a flush an extra 5% of the time, so why does it only win an extra 3.4% of the time? After all, when it makes a flush it will almost always win, and the chance that it is up against a full house or quads is much too small to account for this difference.


Supporting Calculations

You don't need to understand the following calculations or simulations to solve the problem. They simply support the numbers given above. All of the numbers I've given and statements I've made above are accurate, so the problem is not with them or the following calculations or simulations.

The probability that A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/spade.gif[/img] makes a flush against T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] comes from:

[C(11,3)*C(37,2) + C(11,4)*37 + C(11,5)] / C(48,5) =~ 7.2%.

This is the sum of the probabilities of 3,4 or 5 flush cards on the board. The probability that A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/club.gif[/img] makes a flush against T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] comes from:

2*[C(12,4)*36 + C(12,5)] / C(48,5) =~ 2.2%.

This is the sum of the probabilities of 4 or 5 flush cards on the board, times 2 different suits.

Twodimes heads-up result for A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/club.gif[/img] vs. T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img]

Twodimes heads-up result for A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/spade.gif[/img] vs. T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img]

[punter11235]

<font color="white"> Because many times when AKs makes a flush by the river it would win anyway because other A or K is on the board, so those times dont add additional equity </font>
I asked this question to myself some time ago too [img]/images/graemlins/wink.gif[/img]

[onnel]

My off-hand assumption would be that the missing 1.6% are made up of the times that the TT hand either makes quads or a full house and therefore beats the flush.

It's onla a paradox if one assumes that having an A-high flush guarantees you a win over all hands that TT might end up with.

Onnel

[Copernicus]

[ QUOTE ]
Someone asked me a question today that really had me stumped for awhile until I realized what was going on was very simple. I wonder how many of you will be similarly stumped. Pzhon, Aaron, Siegmund and their ilk, let others have a chance to answer.


Problem

A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/club.gif[/img] beats T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] about 42.6% of the time. A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/spade.gif[/img] beats T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] about 46.0% of the time. So being suited increases AK's chance of winning by 3.4%. The advantage of being suited is that the AKs makes a flush by the river about 7.2% of the time, compared to AKo which makes a flush about 2.2% of the time. AKs makes a flush an extra 5% of the time, so why does it only win an extra 3.4% of the time? After all, when it makes a flush it will almost always win, and the chance that it is up against a full house or quads is much too small to account for this difference.


Supporting Calculations

You don't need to understand the following calculations or simulations to solve the problem. They simply support the numbers given above. All of the numbers I've given and statements I've made above are accurate, so the problem is not with them or the following calculations or simulations.

The probability that A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/spade.gif[/img] makes a flush against T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] comes from:

[C(11,3)*C(37,2) + C(11,4)*37 + C(11,5)] / C(48,5) =~ 7.2%.

This is the sum of the probabilities of 3,4 or 5 flush cards on the board. The probability that A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/club.gif[/img] makes a flush against T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img] comes from:

2*[C(12,4)*36 + C(12,5)] / C(48,5) =~ 2.2%.

This is the sum of the probabilities of 4 or 5 flush cards on the board, times 2 different suits.

Twodimes heads-up result for A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/club.gif[/img] vs. T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img]

Twodimes heads-up result for A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/spade.gif[/img] vs. T [img]/images/graemlins/heart.gif[/img]T [img]/images/graemlins/diamond.gif[/img]

[/ QUOTE ]

There are two possible flushes for AKo and only 1 likely for AKs?

Edit: nope your 2* takes care of that

Reedit, and this is probably it without doing the math:

Ties...some of the hands that would have been ties with AKo are no longer ties with AKs, so each of those takes away 1/2 an "Extra" suited win.

[BogusPomp]

I suspect there is some double counting happening.

Say you hold AsKc, and the board comes with 4 spades... one of which is the Ks. Now you hold a pair of kings "or" a spade flush to beat the opponents ThTd. (Assuming the Tens do not improve)?

[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.

[BruceZ]

[ QUOTE ]
Ties...some of the hands that would have been ties with AKo are no longer ties with AKs, so each of those takes away 1/2 an "Extra" suited win.

[/ QUOTE ]

That wouldn't take away 1/2 a win from the suited hand; it would add 1/2 a win to it. If a tie becomes a win when AK becomes suited, that is included in the extra 5% flush wins. So that isn't it. The answer has been given elsewhere in this thread.

[TurtlePiss]

There's a greater chance of the TT making a winning flush against the suited AK if that has anything to do with it. If not, just ignore the previous statement.