Thursday, October 20, 2005

So I got bored, and motivated by my recent hand where I flopped quads with three on the flop for what I believe is the first time in my live poker playing career, I ran some numbers. (These are all "back of the envelope" calculations, so there may be errors. I think my calculations are good, but let me know if you spot any errors.)

The chances of flopping quads given two non-paired cards is (3/25)(2/49)(1/48) = 6/58,800 = 1/9800. That tells me I almost have to have done it before. If so, I really can't remember. I did it in Omaha once that I can remember, but those odds are much better--twice as likely, in fact, at 1/4900. (For those curious, the chances of flopping quads in hold’em given a pocket pair: ~1/408.)

Of course, maybe that's why I remembered that I haven't done it, because even though it's unlikely, the chances of my going whatever high number of live hands I've played without ever having done it is actually a somewhat low 1/351 (approximately, according to this calculator, with N = 80,000, p = 0.00010204 and t=0).

The most statisitically unlikely things that I've noticed happen to me in live play are probably:

1) flopping a royal, 1/649,740, of course (that's pre-deal odds).

2) I got dealt 88 three hands in a row. The odds of getting dealt the same PP three times in a row is 1/830,297.

3) This just happened a few days ago and I wouldn't have believed it if I didn't see it. I got dealt the exact same two cards three hands in a row. Now, I've been dealt the same two cards two hands in a row many times--that carries a probability of 1/1326, so it happens every week or two. The probability of three times in a row? 1/1,758,256. Not only that, but miraculously the two cards were in the exact same order each of the three times. Td on the left, 6s on the right (alas, no Ad As for me). Updated probability: 1/7,033,104. Some people use their luck to win the lottery, I get the Td 6s powerhouse three times in a row.

(And before some wise-guy comes on here and says something, yes I know the chances of my being dealt the exact two cards I've been dealt each hand of my 80,000 live hands is one in eleventy bajillion.)

Other odds and ends:

* I just realized from one of the above calculations that 1/98 = .01020408163265..., that is, it just about lists the powers of two in order (darn 65 instead of 64!). Pretty nifty.

* I've been a baseball fan all my life, and my current favorite team, the Astros, just won the NLCS to advance to the World Series last night. Later on that night, I'm playing poker and this guy asks, "so when does the World Series start?" and the first thing I think is, "dude, the World Series was months ago! They're already showing episodes on ESPN!" Yeah, poker really has taken over my life...

* I'm going back to Austin on Friday to visit, so it might be a week or so before I'm able to update again, depending on how the games go tonight and whether or not I just have to write about it tomorrow morning.


Anonymous Peter B said...


Your "never flopped four of a kind with three on the board" caused me a few minutes' thought.

Clearly your numbers are correct (actually, I just checked them -- they are). And I thought to myself. Hell, it's never happened to me, either. (I've never flopped a royal flush, either, although I did hit my first ever one on the river last March in the Flamingo - and I have seen one player hit two royals within 10 hands).

Anyway, the answer soon became clear. If I play an average of 5,000 hands a month, I MUST have flopped four of a kind with one of the cards in my hand at least once in the past year.

And, of course, I had. I had J7 and the board came 777, just a cuople of weeks ago (the wonders of Pokertracker!). But, equally obviously, I had folded the J7 and not even noticed the board.

If you play tight-aggressive in a tight-aggressive game, a good proportion of the hands that you play will be pairs. Equally clearly, in a tight aggressive game where a flop is seen, it is more likely that one of your cards will be held by another player seeing the flop (thus making it impossible to flop quads).

In other words, hands that you play, in pots where your opponents also play, are in the set where you are less likely to flop quads holding just one of the cards in your hand. You are considerably more likely to flop quads holding a pocket pair.

End of conundrum :-)


10:29 AM  
Blogger eric said...

Ah, thanks Peter, my numbers might be technically correct, but I screwed up the logic in my flopped quads analysis.

It's 1/9800 hands, but like you said I'm not playing every hand. I'm playing 20% of my hands. So instead of 80,000 hands, I really should be doing that calculation based on 16,000 hands, since I have no shot of flopping quads on a hand I folded.

That chances the probability of having never done it to about 19.5%, or about 1/5, which isn't all that spectacular.

And you're right. If I have AK and three other people are in the pot, they're all in the pot because it's quite likely at least one of them also has an A or a K. I'm sure there's some time I raised with AK preflop and the flop would have come AAA or KKK but darnit everyone folded.

I should play more 4/8 to alleviate that problem.

5:18 PM  
Blogger April said...

How long you going be in Austin?

7:26 PM  
Blogger eric said...

I'll be in Austin Fri-Thur. Good place to be while the Stros are in the series.

It's been a while since I've been somewhere that people care about the outcome of a series for reasons other than the money involved.

12:05 AM  
Anonymous Greg P said...

I don't remember if I've flopped quads that way before, but I have had a few statisitically unlikely things happen while dealing.

1. The first TWO straight flushes I dealt were both 5-9, both spades, and both were dealt to players in the five seat on table ten.

2. The first royal I dealt was on the board, and again in spades. TsQsAs on the flop, Ks on the turn, Js on river. It was in a 3-6 holdem game too, and the players didn't even know it was an auto-split.

Also, Eric, I think I dealt to you this past sunday. Meant to come by afterwards and ask if it was you, but it was late for me, and I forgot. Didn't really know what you look like, but was just looking for someone arriving late to the 80-160, playing well, and looking "like a blogger." Anyway, the guy was wearing glasses and a "TEXAS LAW" t-shirt, and in the six seat on table 27, the one on the lower level near the high limit terrace; sound like it could've been you?

5:31 AM  
Anonymous Evan said...


That definitely sounds like Eric. Was his head shaved or close to it?

7:37 AM  
Blogger eric said...

That was almost definitely me.

Now, how long until the players find this blog and learn how I think and play. hmm...

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