Flopped nut straight %

[Percussion]

First of all, what is the % of AQ flopping KcJsTd (or any rainbow.)

Lets say that that opponents have A2c, A2s, A2d, and one has KK.

How often does this hand stay the nuts?
What is the minimum you can bet on flop if you know these players will call 100% of the time (The backdoor flushes will stop if they miss the turn.)
River?

Pointless or thoughtful question?

[jba]

1- get pokerstove
2- if they're calling 100% of the time, and you have the nuts, anything but going all in is a gigantic mistake.

This is my first odds post, so lets see if I can not screw it up. [img]/images/graemlins/smile.gif[/img]

How often will you flop the nuts, and they will be rainbow:

Well, there are two kings, if you hit one of those, there are 3 Jacks of a different suit, and then 2 tens of different suits than either, so there are 2*3*2 = 12 ways to hit the rainbow straight. We know 10 cards, so there are 42*41*40 = 68880 ways for the flop to come down. Your chance to hit this is 0.000174/1, which is 0.0174% of the time.

Supposing you hit it, how often will it hold up? Well, let's assume you can't hit a flush, and that, as you've indicated, the players with A2 all hit a third card of their suit on the flop. Suppose that between the two kings and the AQ, each player has lost one extra card to his flush. Then there are 9*8=72 ways for any one of these players to make a flush, so there are 72*3=216 ways for the flush to be made against you. Further, there is a king (which we may be double counting, but it's not a huge error) which makes four of a kind, and the jack and ten of the fourth suit make full boats, so that's 3 more cards that will ruin you. There are 38 ways for any of these cards to come out on the turn or the river (that card, and one other). There are 3 ways for two to come out, so there are really only 38*3-3=111 ways for this to be a problem. So, we have 216+111=327 ways for the turn and river to ruin you. There are 39*38=1482 ways for these two cards to come down. So you'll be busted 327/1482 = 22.06% of the time.

Probably I've made mistakes, someone want to point these out for me?

As for how much you should be, I'll leave that to someone else.

--Big Chris

[jba]

BigChris:

you are highly discounting the chance for the kings to catch you. aside from the J or T, the turn card can also pair on the river for him to win. Also, he only needs to catch one card, not running cards like the flushes to win.

actual results:

741 games 0.005 secs 148,200 games/sec

Board: Kc Js Td
Dead:

equity (%) win (%) tie (%)
Hand 1: 43.0837 % 40.62% 02.46% { AhQs }
Hand 2: 06.2416 % 03.78% 02.46% { Ac2c }
Hand 3: 05.2969 % 02.83% 02.46% { As2s }
Hand 4: 06.2416 % 03.78% 02.46% { Ad2d }
Hand 5: 39.1363 % 39.14% 00.00% { KdKh }

Ah, you are right, I hadn't thought about the board pairing at the end - nice catch, thanks for pointing this out. 3s-9s can pair in 6 ways each, for 7*6=42 more ways to lose. Queens can pair in 3 ways, which will also lose for you. We have 45 more ways to lose, so we now lose 372/1482 = 25% of the time.

This isn't nearly as high as your simulation predicts though. In particular, consider the Kings. there are 42 ways for river to pair the turn in 3s-9s and 3 ways for Qs to pair. There are 3 Js and 3 10s left. A J can come out in 3*38-3=111 ways (3 suits, 38 other cards, minus 3 for double counting when 2 Js come out). Same for 10s. Finally, a fourth king can come out in 38 ways, of which we have counted 6 already (KJs and K10s), so 32 more ways to lose. This is 42+3+111*2+32=299 ways to lose to the Kings. This is only 299/1482=20% of the time.

Apparently I'm a making mistake somewhere, but I don't see it - can anyone help me out? I haven't been working with poker odds for very long.

[jba]

[ QUOTE ]

This isn't nearly as high as your simulation predicts though. In particular, consider the Kings. there are 42 ways for river to pair the turn in 3s-9s and 3 ways for Qs to pair. There are 3 Js and 3 10s left. A J can come out in 3*38-3=111 ways (3 suits, 38 other cards, minus 3 for double counting when 2 Js come out). Same for 10s. Finally, a fourth king can come out in 38 ways, of which we have counted 6 already (KJs and K10s), so 32 more ways to lose. This is 42+3+111*2+32=299 ways to lose to the Kings. This is only 299/1482=20% of the time.


[/ QUOTE ]

you are pretty close. when you say there are 3 ways for the river to pair Qs the math you are doing is C(3,2) which is 3*2/2!. the division by 2! is because we do not care about the order of the cards. When you calculate the total number of turn/river combos you also must divide by 2! for the same reason, 299/741=0.403508772 which is closer. I think you also double counted the instances where the turn and river are J and T which is 3 ways, which takes us down to 296, but the numbers indicate we're still double counting somewhere.

I would approach this problem differently. There are 39 unknown cards on the turn; of those, 7 cards give the kings a winner, 1 card (case deuce) gives him 7 outs on the river, 3 cards (Qs) give him 9 outs on the river, and 28 cards give him 10 outs on the river.

the ways that he *misses* are:

deuce and a blank: 1/39*31/38
queen and a blank: 3/39*29/38
two total blanks: 28/39*28/38

he can miss any of these non mutually exclusive ways, so the chance that he misses is 1/39*31/38+3/39*29/38+28/39*28/38, and the chance he sucks out on you is:

1-(1/39*31/38+3/39*29/38+28/39*28/38)=0.391363023

[ QUOTE ]
Pointless or thoughtful question?

[/ QUOTE ]

I don't think the situation will occur often enough for you to really worry about it. ie not very much practical benefit to having this knowledge, but it could be interesting from a mathematical perspective. I guess that's what the theory board is for.

[jba]

[ QUOTE ]
[ QUOTE ]
Pointless or thoughtful question?

[/ QUOTE ]

I don't think the situation will occur often enough for you to really worry about it. ie not very much practical benefit to having this knowledge, but it could be interesting from a mathematical perspective. I guess that's what the theory board is for.

[/ QUOTE ]

tell me please how there's not much practical benefit to knowing the difference between having 20% and 40% equity in a gigantic multiway pot.

Thanks very much, I definitely forgot ignore order in those cases, and am terrible at spotting my own mistakes. Your method is also very nice, and I will probably employ it in the future.

jba: I'm referring to the parlay of events suggested by OP. Getting AQ AND flopping the nut straight in a 5 way pot with these specific hands likely playing for 2 or 3 bets each AND have all of them play to the river, in the face of excessive action from KK and AQ.