Part 1. Your flush questions addressed.
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Suppose I have Ad2d3hKh in a 10-handed game.
Somewhere I have seen a stat that says that there is a 36% chance that someone was dealt the nut heart draw.
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Jim - Before you see the flop, you know the whereabouts of 4 cards and the whereabouts of the other 48 are unknown.
When you see the five card board on the river, you know the whereabouts of 9 cards and the whereabouts of the other 43 are unknown.
Before the flop, the probability one of your nine opponents was dealt the ace of hearts is 36/48. After all five board cards are known, and if the ace of hearts is not one of them, the probability one of you nine opponents was dealt the ace of hearts is 36/43. Those probabilities are simply the number of cards your opponents have collectively been dealt divided by the total number of cards possible.
Assuming one of your opponents was dealt the ace of heart, the probability that opponent was also dealt another heart has three cases. (1) one other heart, (2) two other hearts, and (3) three other hearts. All three must be considered and then combined (by adding).
Then the probability an opponent was dealt the ace of hearts is multiplied by the probability that opponent was dealt one, two, or three other hearts. I’m sure I waded through that at one time or another and posted the result both here on 2+2 and on r.g.p. I would have been more interested in nine handed than ten handed, but I imagine I computed it for both.
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However, I am not so sure this number is accurate
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And you still won’t be sure if I wade through the calculations again.
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The only time I care is when I see the board.
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Fair enough. In addition to the only time you care, I think I might have been interested in knowing how often a king high flush draw might run into an ace high flush draw, figuring from immediately after a flop that had two cards of the flush suit. There was some crap that kept getting posted about never drawing to anything but the nut flush. (I’m not advising anybody to draw to anything but the nut flush, but I do think non-nut flush draws, especially to the king, do add value).
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But now the deck composition is very different. The pool of unseen cards is no longer 48 cards with 11 hearts.
Assuming we are at the river, the pool of unseen cards is down to 43 and only 8 of them are hearts.
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Good thinking. Which ever one you’re interested in, they clearly are two distinct problems - or three distinct problems if figured from immediately after the flop - or four distinct problems if you want to throw in the turn.
But if there are two distinct problems, then there are also two distinct answers.
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It seems this makes the K-high flush quite a bit more likely to be the winner. On the other hand, I don't know how the 36% figure was calculated or if it accounted for the board cards in this way.
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Tell you what. I’ll run a ten handed simulation, giving Hero
K[img]/images/graemlins/heart.gif[/img], K[img]/images/graemlins/club.gif[/img], 2[img]/images/graemlins/heart.gif[/img], 3[img]/images/graemlins/diamond.gif[/img], and making the board on the river
<font color="white">_</font>
Q[img]/images/graemlins/heart.gif[/img], 8[img]/images/graemlins/heart.gif[/img], 7[img]/images/graemlins/heart.gif[/img], J[img]/images/graemlins/club.gif[/img], T[img]/images/graemlins/diamond.gif[/img]. The only way Hero’s hand can be beaten is by an ace flush.
10,000 deals good enough? That should get us reasonably close to the truth. I’m predicting Hero’s hand will win about 64% of the time.
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6389 wins and 3611 losses. Well... that rounds off to 64%.
Next I’ll run another ten handed simulation, giving Hero
K[img]/images/graemlins/heart.gif[/img], K[img]/images/graemlins/club.gif[/img], 3[img]/images/graemlins/heart.gif[/img], 2[img]/images/graemlins/diamond.gif[/img], and making the flop Q[img]/images/graemlins/heart.gif[/img], 7[img]/images/graemlins/heart.gif[/img], 2[img]/images/graemlins/heart.gif[/img]. This time Hero will still probably end up with the second nut flush on the river, but this time the board may pair on the turn or river. In addition, 44/990, Hero figures to end up with the nut flush on the river. Also, there’s a very remote chance of an opponent ending up with a straight flush. Finally, note that Hero could win with a full house or quads. Will hero win more or less than in the last sim?
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Ready?
After this flop, Hero has 4291 wins for flush, 242 wins with full house, 15 wins with quad kings. 4548 total wins.
Hero has 181 losing full houses, 5271 losing flushes.
Hero loses with a flush more than he wins, but more to full houses than to ace-high flushes.
Opponents collectively have 5597 wins.
• 12 winning straight flushes,
• 301 winning quads,
• winning full houses 2985 times, and
• 2154 winning flushes.
The simulator doesn’t provide a way to know how often an opponent had an ace-high flush that was pre-empted by a full house. (Or if there is a way, I don’t know how).
But since there were two more non-hearts, Hero’s opponents figure to have a heart flush slightly less often than on the river. kind of a moot point when you notice how often Hero gets beaten after the board pairs. Getting beaten by the board pairing and an opponent making a full house or quads is more prevalent than getting beaten by a better flush by about a three to two ratio.
At any rate, Hero wins more with the second nut flush on the river than with the second nut flush on the flop. That seems entirely reasonable.
Let’s back it up to before the flop. We’ll give Hero K[img]/images/graemlins/heart.gif[/img], K[img]/images/graemlins/club.gif[/img], 3[img]/images/graemlins/heart.gif[/img], 2[img]/images/graemlins/diamond.gif[/img] again but leave the board blank. ten handed, 10,000 runs. Of course this time Hero can make a full house and a straight, but is not guaranteed a flush.
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This time Hero wins more with a full house or quads (607+113) than he wins with a flush (294/10000) and Hero loses more with the flush (369/10000) than he wins with it. But with 23KKs, the hand wins for low more than for high and the suited king just changes some of those low wins to scoops and some of those low quarters to three quarters. You’re not really playing the hand exclusively for the suited king, but rather for a combination of different potential ways to win which include the suited king.
Bear in mind that these sims involve non-folding opponents. In a real game you have to adjust for folding opponents.
Buzz
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