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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]
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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.