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So I've been trying to figure out the probability of flopping the unbreakable nuts. And I'm a little confused on how exactly it will work out. Essentially, I think that having some sort of straight could possibly be the unbreakable nuts but then again, full house, flush, four of a king, straight flush are so much higher than it.

I don't know how to approach this problem. Any help would be highly appreciated!

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    there's no such a thing as unbreakable in poker, unless you're talking about Royal Flush in which no-one would dare to play anyway (action freeze). Even flopped quads can be beaten by higher quads till River, mathematically. Is this just a math problem? Odds for unbreakable Royal Flush? about 0.0000zzz. – user1165 Jan 27 '15 at 6:38
  • Yeah, its a probability problem. And that's what I was thinking too that it could only be like royal flush but then again, the unbreakable nuts with your hand and the flop, could be like two pair and no one has higher than that, you know? Not sure if I'm thinking of this in the right direction though – user2582622 Jan 27 '15 at 6:42
  • the exact percentage is about 1/650000 ;) No, a 2-pair can be cracked by a higher 2-pair or set that comes to Turn or River, assuming the board is dry that doesn't allows any flush/straight possibilities. – user1165 Jan 27 '15 at 6:50
  • Is that just the probability of getting a royal flush? – user2582622 Jan 27 '15 at 6:59
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    Even if you find the correct number, it will be so small that it will have absolutely no relevane in practice. I'm curious: why would you want to know this ? – Radu Murzea♦ Jan 27 '15 at 7:55
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a royal flush is not the only "unbreakable nuts".

sometimes, a straight flush is unbreakable.

example: if the flop is 5d, 7d, 8d and i am holding the 6d and 9d, my hand is unbeatable regardless of what other cards come out and what other players hold.

it is important to remember there is no difference than a straight flush and a royal flush except that the royal flush is the highest straight flush there is.

also, the only hands that are unbreakable at the flop are straight flushes, and you must be holding the highest card of the straight flush in your hand.

there is no way to eliminate the possibility of an eventual straight flush. no matter what 3 cards fall, 2 additional cards and 2 cards in an opponent's hand could make a straight flush... making even a flopped 4 of a kind in Aces beatable.

therefore, i believe to find the odds of an unbeatable hand on the flop to simply take the odds of flopping a royal flush, and multiply that by 2/5 (the odds of holding the top card of the flopped straight flush). i could be wrong, but i believe that would give you the answer you are looking for.

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    <pedantic> You don't always have to have the top card in a straight flush to make it unbreakable. For example, if you have T8s and the flop comes J97 of your suit, you're unbeatable, since there aren't enough cards on the high end to make a higher straight flush than yours. – Chris Farmer Jan 27 '15 at 18:34
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    you are correct chris. i failed to account for that. i suppose anytime you are holding a Ten or higher in a flopped straight flush, you are now unbeatable. – sakau2007 Jan 27 '15 at 18:35
  • AK vs KK and flop is AAx rainbow (x is not T/J/Q) ? – OldPadawan Jul 1 '17 at 5:07
  • @OldPadawan In determining whether a straight flush could fall by the river you can only consider the board, not the hole cards. with AAx (x not T/J/Q) the turn and river could for instance be K and Q of matching suit to one of the ace so JT of the same suit in the hole would make a straight flush. – user1934 Jul 1 '17 at 16:46
  • My comment is not about str8 flush of course, as OP asked for unbreakable nuts, answer and comments already mentioned str8/royal, I was wondering if AK vs KK and flop AAx would be unbreakable, and therefore, help... I misunderstood ? – OldPadawan Jul 1 '17 at 16:52
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Any straight flush where you hold the T-A as you have blocker to the royal

xx is hole cards
yyy is board

nut
xxyyy

xyxyy

xyyxy

xyyyx

not nut
yxxyy *
yyxxy
yyyxx

yxyxy *
yyxyx

yxyyx *

`* near statistical nut as you are 178364 : 1

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