Wednesday, August 12, 2009


Pocket aces, again!Was scanning through the Two Plus Two forums yesterday when I noticed a thread that had begun during the spring that got “bumped” to the front page thanks to a new post or three. Not really that interesting of a thread, and in fact it looks like the mods swiftly closed it as there didn’t seem to be that much left to say about the original topic.

The thread appeared in the “Two Plus Two Magazine” forum and was titled “How is being delt [sic] aces twice in a row not over 40,000-to-1?” The original poster’s question was in response to something Mason Malmuth had written in his “Publisher’s Note” to the April 2009 issue of the online magazine.

At the end of his note, Malmuth mentioned how he’d been playing limit hold’em at the Bellagio and a woman had been dealt pocket aces on two consecutive hands. “She couldn’t believe her good fortune and wanted to know what the odds of that happening were,” tells Malmuth. “Another player spoke up and told her that it was over 40,000-to-1. Of course this is wrong.”

The subsequent debate revealed that, in fact, “over 40,000-to-1” does correctly describe the odds of getting dealt A-A on two consecutive hands -- i.e., (1/221)(1/221) = 1/48,841. Of course, the odds of getting dealt pocket aces on the hand after the one in which you just got pocket aces is still just 220-to-1, as it always is for every hold’em hand. Indeed, Malmuth himself chimes in on the first page of the thread to say as much.

As I say, not that interesting of a thread, and it seems like the only real debate is over accuracy in one’s phrasing. As we all know, the odds of being dealt any two cards are entirely unrelated to whatever cards one was dealt before. As Malmuth, Ed Miller, and David Sklansky say in one of my favorite passages in Small Stakes Hold’em (2004), “each hand is an independent event. Cards are pieces of plastic. They have no knowledge, no memory, no cosmic plan. They are scrambled and shuffled thoroughly prior to every hand. Pieces of plastic cannot possibly conspire against you....”

'Poker Nation' by Andy Bellin (2002)In Andy Bellin’s Poker Nation (2002), he relates the story of a small-time pro named Dicky Horvath. He quotes Horvath describing how boring the lifestyle of the small-time grinder can be. “‘The monotony is what kills you,’” says Horvath. “‘Not the gambling. You got to remember that poker is a finite game. There are only so many variations. I’ve been dealt pocket aces five hands in a row. That’s like a billion to one odds. I figured it out once.”

Some may want to quibble with the precision of Horvath’s calculation. But really, who cares? Getting aces twice in a row is quite rare. Five times is extraordinarily rare. The odds of either are not really worth knowing, other than to satisfy our curiosity for mathematical trivia.

For some reason, thinking about this whole issue of getting dealt pocket aces twice put me in mind of baseball and its many statistics and records. As a kid, I was a fanatic, and like many young American boys memorized all of the important numbers associated with baseball records -- numbers like 4,191, 56, 755, .367, 61, and 511. Some of the records represented by those numbers have since been broken. Some haven’t yet.

They say all records are made to be broken, but there’s one baseball record that I think it is safe to say will probably never be broken. And it sort of resembles getting dealt pocket aces twice in a row.

Johnny Vander MeerI’m referring to Johnny Vander Meer’s having thrown no-hitters on two consecutive starts (June 11, 1938 and June 15, 1938). Someone one day might also get two, but no one is going to get three, I don’t think.

That’s like a billion to one odds. I figured it out once.

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Blogger dbcooper said...

I was in Vegas playing at a 1/2NL game and saw a guy at our table get pocket aces 3 hands out of 4. They held up twice.

8/12/2009 10:54 AM  
Anonymous Dave said...

I agree to agree with you about the baseball record. But apart from that, isn't it nice to have some baseball records which are still sacrosanct.

8/12/2009 11:24 AM  
Blogger Unknown said...

Nice write!

8/12/2009 1:17 PM  
Blogger gadzooks64 said...

I got aces 3/5 hands on Full Tilt a while back but that's not so unexpected because we all know online poker is rigged ;)

During my last live session the gentleman on my left picked up aces twice inn a short amount of time AND he flopped quads with them BOTH times.

I don't even want to know the odds of that happening twice in one session. Live poker is so rigged.

Ok, I think we can all agree that poker is just plain rigged. Amen.

8/12/2009 3:08 PM  
Blogger Greylocks said...

I'm probably repeating something that was said in the thread, but I'll say it anyway.

The probability of being dealt a specific sequence of consecutive hands depends on the size of the sample.

For example, the probability of being dealt AA two times in a row in a sample of two hands is 1/48,841, but the probability of it happening at least once in a sample of 10 is about 1/22, in a sample of 100 hands about 1/2.7.

In a four-hour session of reasonably fast live Holdem you'll probably get dealt around 160 hands, in which case there's roughly a 50% chance you'll get dealt aces twice in a row that session.

In other words, getting dealt the same type of hand twice in a row is actually pretty common in Hold'em, far more so than people seem to think. I think if people were to actually keep track of how often they get back-to-back aces in Hold'em, they'd realize it's happening more often than they perceive.

BYW, I got dealt quads (8's) in PLO today. The probability of any specific starting Omaha hand being any four-of-a kind is 1/20825. But I play anywhere from 6,000 to 10,000 hands of PLO or PLO8 a week so I'm going to get dealt quads a few times a year.

8/14/2009 11:02 PM  

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