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Phil Pilgrim (PhiPi)
02-12-2012, 11:40 PM
I play poker with a group of friends on a fairly regular basis. We've been playing together for more than thirty years. In that time, we've seen some odd runs of the cards, but nothing like what transpired in our most recent get-together. One of the games we play involves dealing out cards two-at-a-time, face up. This time, however, the dealer dealt out six pairs in a row! The obvious question was, "What are the odds of that ever happening?" So, being the computer nerd of the group, I was elected to compute those odds.

In order to simplify the problem somewhat, I've restated it as, "What is the probability that, starting from a shuffled deck of 52 cards, and dealing two cards at a time from the top, that the first six duets will be consist of six different pairs?" We can start by computing the probability for the first pair. Since each rank (ace through king) has four suits, there are six distinct and equally-probable pairs that can be formed from them. This is given by the formula for combinations of n things, taken m at a time:


C(n, m) = n! / [m! (n - m)!)]

So the total number of first pairs from all 52 cards is 13 * 6, or 78. The total number of distinct two-card combinations, pairs and non-pairs included, is


C(52, 2) = 52! / (2! 50!) = 52 * 51 / (2 * 1) = 1326

So the probability that the first two cards dealt out will be a pair is


78 / 1326 = 0.0588235 or 5.88%

For the second duet, assuming the first was a pair, the of ranks available to form a different pair now number 12, and the number of cards remaining is 50. So the probability that it will be a pair is:


12 * 6 / [50! / (2! 48!)] = 72 / 1225 = .05877551

The probability that the first two duets will be different pairs is the product of the two probabilities computed thus far, or .00345738.

We can continue in this fashion for the next four duets dealt and arrive at the final answer:


(78 / 1326) * (72 / 1225) * (66 / 1128) * (60 * 1036) * (54 / 946) * (48 / 861) = 3.72833E-8

That's one chance in 26.8 million. I don't think we'll ever see that happen again.

-Phil

ElectricAye
02-12-2012, 11:54 PM
...
That's one chance in 26.8 million. I don't think we'll ever see that happen again.

...

What might be equally interesting is trying to compute the odds of something happening at a poker game that causes everyone to gasp, "Geez, what are the odds of that ever happening?"

Phil Pilgrim (PhiPi)
02-13-2012, 12:46 AM
What might be equally interesting is trying to compute the odds of something happening at a poker game that causes everyone to gasp, "Geez, what are the odds of that ever happening?"
No computation necessary: it's at least 99.999%. :)

-Phil

jmg
02-13-2012, 04:21 AM
"What is the probability that, starting from a shuffled deck of 52 cards, and dealing two cards at a time from the top, that the first six duets will be consist of six different pairs?"

I get a slightly different answer :)

(1*3/51) * (48/50)*3/49 * (44/48)*3/47 * (40/46)*3/45 * (36/44)*3/43 * (32/42)*3/41 = 3.7319338077e-8

Cluso99
02-13-2012, 09:49 AM
Forget the odds and buy a lotto ticket or whatever name they go under in the USA.

Mark_T
02-13-2012, 04:48 PM
Those are the odds assuming a perfectly random shuffle - I doubt this was actually the case (you need 226 bits of entropy at least to do that) http://en.wikipedia.org/wiki/Shuffling#Randomization

Phil Pilgrim (PhiPi)
02-13-2012, 04:56 PM
Before the cards were dealt, they were subject to a wash shuffle, then to several riffle shuffles, followed by an overhand shuffle. They were thoroughly mixed.

-Phil

erco
02-13-2012, 08:21 PM
Poker night with PhiPi's friends? A higher collective IQ than yer average Jack-swillin', poker playin' crowd, I'll wager.

Q: What has an IQ of 150 and a full set of teeth?
A: The entire front row at Wrestlemania!

idbruce
02-13-2012, 08:27 PM
Does the entire hand consist of only two cards? If not, did the hand with the initial highest pair win the pot and how did the betting transpire?

Phil Pilgrim (PhiPi)
02-13-2012, 09:14 PM
One card each is dealt face-down to everyone. Then the player to the dealer's left is offered two cards from the deck, face up. He can accept or refuse them. If he refuses, they're offered to the next person. If he also refuses, the next person has to take them. Then, there's a round of betting, after which the process is repeated beginning with the player two to the left of the dealer. Another round of betting, offers, etc., rotating until all the players who haven't folded have seven cards. Then one final round of betting. The game can be played high, low, or high-low-split.

It's a stupid game that I detest, BTW, and I seldom stay in to the end. Due to my protests -- "I hate this ****ing game!" -- whenever the dealer calls it, it's been named after me among our group. It's been that way so long, I've forgotten what its real name is.

-Phil

idbruce
02-13-2012, 09:40 PM
Phil

Yea, I like straight poker myself. When I lived in Vegas the first time, I was a regular at the seven card stud tables. Then when I lived Vegas for the second time, approximately 15 years later, most of the casinos had done away with seven card stud. At which point, I had to learn Hold'em, which is okay I guess, but I still prefer seven card stud or five card draw. The only other gambling card game that I really like is a good old fashion game of "In Between", because I always get lucky :)

Bruce

erco
02-13-2012, 10:06 PM
I think televised poker a few years back is one of the reasons I stopped watching TV. It proved we'd bottomed out as a society, and that there truly was nothing good on TV.

DavidSmith
02-14-2012, 01:18 AM
In 1970, one of my professors demonstrated that it is functionally impossible to have a finite string of numbers in which a pattern can not be found.

So, if a rare event is anything w a pattern, then the odds are pretty close to 100%.

Phil Pilgrim (PhiPi)
02-14-2012, 01:53 AM
In 1970, one of my professors demonstrated that it is functionally impossible to have a finite string of numbers in which a pattern can not be found.
As a last resort, there is always a LaGrange polynomial available to produce them! :)

-Phil

DavidSmith
02-14-2012, 02:07 AM
True, but it was a class in statistics and continuous functions were not being used.

The specific example the prof gave (if anyone is really interested) is that the "random" drawing of draft numbers for the Viet Nam war had just taken place and someone had noticed there was a preponderance of larger numbers in the first half of the drawing. OBVIOUSLY the balls had been put in in numeric order and insufficiently mixed.

Even though I was taking the class and approaching my degree, I don't think I really understood the logic/math. (Some of the professors at UCLA were something else. We had one guy (thermodynamics) who was just plain slow. Ask him a question and he would rare back and think and hem and haw for a while. But, once he got started the partial differential equations of the answer would haunt your dreams that night).

Phil Pilgrim (PhiPi)
02-14-2012, 02:38 AM
The specific example the prof gave (if anyone is really interested) is that the "random" drawing of draft numbers for the Viet Nam war had just taken place and someone had noticed there was a preponderance of larger numbers in the first half of the drawing. OBVIOUSLY the balls had been put in in numeric order and insufficiently mixed.
That's not how the drawing worked. What was drawn were birthdates, not draft numbers. The draft numbers corresponded to the order in which the birthdates were drawn. It's an evening I shall never forget. I returned to my dorm room after the first drawing had commenced (broadcast live over the radio). It was in the 60's when I started listening, and it got up into the 300's without my birthdate having been announced. I was certain, by then, that I had a low number. But I lucked out: 332.

Next morning, in my first Physics class of the day, guys were comparing numbers. One announced that he had the highest number: 365. Another countered, "I've got you beat: 366!"

I hope, for the sake of my nephews, that we never see another time like that. I'm grateful for the service of our volunteers, but no one should be coerced to serve against their will.

-Phil

Loopy Byteloose
02-14-2012, 12:46 PM
If totally random, it certainly would be a rare event. But if you study how to cheat at dealing at cards; it is quite an easy trick to achieve.

First, pick up the previously played cards by pairs and put all of them on the bottom of the deck.

Second, shuffle the deck in a manner that avoids reaching the preprepared bottom cards

Third, either deal from the bottom of the deck or manage a cut that positions all the pairs to the top of the deck and ready to deal.

This cheat even gets easier if two people work together as your partner can gather up cards and arrange them while you grab random cards. Then your partner hands you the prepared stack that most trusting people have not observed being prepared.

The simple fact is 'the hand is quicker than the eye' and a practised card cheat dealing from the bottom of the deck is not that hard a skill to acquire.

I gave up on poker as I played bridge in college with a group that worked their summers as dealers in Las Vegas. The first thing the casinos did was to train them in how to cheat so they knew that the pit bosses knew what to watch for.

The only way to avoid dealing off the bottom is to require the dealer to properly grip the deck at all times and to deal right handed - rare in a friendly game.

My Vietnam draft lottery number was #3 and I spent the evening drinking for free in the tavern where we watched the lottery. (Luck be a lady tonight....)

skylight
02-14-2012, 09:17 PM
As a youngster I used to accompany my grandmother to "Whist Drive" evenings playing Whist or otherwise known as Trumps, I was dealt a hand that contained the complete suit that happened to be trumps at the time so an unbeatable hand but more astonishingly another player was also dealt a complete suit.
I remeber getting told off by one of the elderly players for laying my hand down straight away instead of playing them one at a time even though it was an unbeatable hand :smile: