Perfect flop odds

[JDErickson]

Anyone know what the odds of flop coming KKK and 1 player holding the other K? Or any other card 333,3 999,9

Thanx

[Dynasty]

When holding a non-pair hand such as T4o, the odds of flopping quads is 9,799:1.

[ddubois]

Dynasty's reply is misleading. It is extremely rare to flop quads, but once you have the 3-of-a-kind flop as a given, the odds of quads being out there changes dramatically.

The odds of the 4th K being out there are the same as any player holding the 2/images/graemlins/club.gif or the A/images/graemlins/heart.gif or any other singlular individual card. Assuming 9-handed and every player saw the flop, it would be

P(odds of no others holding an individual card) =
46/47*45/46* 44/45*43/44* 42/43*41/42* 40/41*39/40* 39/40*38/39* 37/38*36/37* 35/36*34/35* 33/34*32/33 = .6638

P(someone holding that card) = 1 - P(odds of no others holding an individual card) = 33.6%

It should be easy to generalize to N hands, something like:
1- ( 47!/(47-2n)! / 52!/(52 - 2n)! )
although I'm not positive that formula is correct. College was a long time ago.

One caveat though is that not eveyrone plays all their hands. If the flop is AAA, your more likely than "the average trips flop" to be up agaist quads; if the flop is 222, you're more likely than "the average trips flop" to be up against some mucker muttering to himself "F@#$! I WOULD HAVE HAD QUADS!"

[tripdad]

[ QUOTE ]
When holding a non-pair hand such as T4o, the odds of flopping quads is 9,799:1.

[/ QUOTE ]

so you're saying it's ok to call preflop with T4o?

cheers!

[Lost Wages]

46/47*45/46* 44/45*43/44* 42/43*41/42* 40/41*39/40* 39/40*38/39* 37/38*36/37* 35/36*34/35* 33/34*32/33

Reduces to 32/47 by cancellation which is 0.681 or
P(someone holding that card) = 31.9%

Lost Wages

[JDErickson]

WOW. Thanx for all the info. I would have thought the chances someone having the case card would be a lot less. Guess not.

This was the hand that started the question.

99 UTG and I raise. BB calls. Flop KKK. I bet he calls. Turn 7, I bet he raises, I call. River 4. I ck call. He has KQo

Was the call down justified????

[ddubois]

Here you have 1 caller, not a full table. It's possible the king was dealt, but folded preflop as part of a K4o hand. The odds of any one random hand having the case K is much different... = 1 - 45/47, or roughly 4%. Then again, 4% is too conservative, because people play hands with Ks more often than random hands.

I can't say whether you should have called down. He's certainly representing a big hand on the turn with the check-raise, but lots of scenarios are reasonable: 1) he could have a smaller PP, 2) he could have a 7, like A7, 3) he could be bluffing, 4) he could be semi-bluffing something like AJ, 5) he could have a larger PP than 99, 6) he could have the case K. Calling down seems reasonable I suppose.

[BugsBunny]

I think calling down here is fine. He could be on a lot of things here some of which you beat, some of which you don't. If your pp was smaller than 7's I'd be more likely to fold here, but even then I think I'd usually call down (player dependent).

[bisonbison]

It'd be fun to know what percentage of the time any individual rank is played. If a king is dealt to a player, how much more likely is it to be played than if a six is dealt, independent of the other hole card?

We all feel more threatened when paired queens flop as opposed to paired fours, but how much more likely to see the flop is the Qx vs. 4x?

[ddubois]

[ QUOTE ]
how much more likely to see the flop is the Qx vs. 4x?

[/ QUOTE ]

Yes, that is an interesting question. I think PokerTracker is just a front end to an access database, so there's probably some way to extract this.

Instead, I just whipped up a chart, and made up what I thought might be reasonable seen-flop-% values for each hand. I tried to take into account you see the flop with every hand sometimes because of free plays in the blinds. Ok, these numbers are probably very wrong in places, but you get the idea:

<font class="small">Code:</font><hr /><pre>
Hands Ways SF%
QAs 4 95 3.8
QAo 12 90 10.8
QKs 4 90 3.6
QKo 12 80 9.6
QQ 6 95 5.7
QJs 4 80 3.2
QJo 12 50 6
QTs 4 75 3
QTo 12 40 4.8
Q9s 4 50 2
Q9o 12 5 0.6
Q8s 4 5 0.2
Q8o 12 5 0.6
Q7s 4 5 0.2
Q7o 12 5 0.6
Q6s 4 5 0.2
Q6o 12 5 0.6
Q5s 4 5 0.2
Q5o 12 5 0.6
Q4s 4 5 0.2
Q4o 12 5 0.6
Q3s 4 5 0.2
Q3o 12 5 0.6
Q2s 4 5 0.2
Q2o 12 5 0.6
58.7

4As 4 30 1.2
4Ao 12 15 1.8
4Ks 4 10 0.4
4Ko 12 5 0.6
4Qs 4 5 0.2
4Qo 12 5 0.6
4Js 4 5 0.2
4Jo 12 5 0.6
4Ts 4 5 0.2
4To 12 5 0.6
49s 4 5 0.2
49o 12 5 0.6
48s 4 5 0.2
48o 12 5 0.6
47s 4 5 0.2
47o 12 5 0.6
46s 4 5 0.2
46o 12 5 0.6
45s 4 15 0.6
45o 12 5 0.6
44 6 70 4.2
43s 4 10 0.4
43o 12 5 0.6
42s 4 5 0.2
42o 12 5 0.6
16.8
</pre><hr />

So according to this, I would probably see the flop with Q* roughly four times as often as I see a flop with 4*.