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Apr 23 2013

Last poker hand in Casino Royale

I was intrigued following my review of Skyfall when commenter Enkidum said that the implausibilties in this film pale “in comparison to the final poker hand in Casino Royale, which has never occurred in history and likely never would, even if people kept playing poker until the heat death of the universe.”

For those who cannot recall that scene, here it is. In this form of poker, five cards are placed face up and are common to all players, while each player has two cards that are unknown to everyone else. The winner is the one who can make the best combination out of the seven cards.

As other commenters pointed out, any hand in poker (or bridge) is as likely as any other, so why do we think of some hands as being ‘rarer’ than others and are surprised when they occur? The answer is of course because we rank order all the possible hands and assign different values to each based on a purely arbitrary system, and choose just a few (those with easily recognizable patterns) to have high value and many to have low value. So, rather than rare hands being high value we make high value hands be rare. But because the creators of the game have chosen easily recognizable patterns to be the high value ones, we are fooled into thinking that such patterns are rarer than other, less obvious, patterns.

In order to increase the dramatic effect in the film, they made the first player have a high value hand in order to bid high, and he reveals that he has a flush, which is pretty high value and thus quite rare, and also tightly restricts the number of hands that the others must have in order to beat it. The filmmakers wanted the next player to have a hand (full house) that is higher than the first but not as high as the subsequent player, which severely restricts the number of possibilities. The third player’s hand had to be such as to beat the second player but not Bond as the last player, squeezing the range of options even more. They gave him an aces high full house and Bond wins with a straight flush.

So while any one of those hands is rare but not unbelievably so, Enkidum is right that that particular sequence of successively higher value hands is extremely rare because each one has to fit into a very narrow spectrum of hands. I haven’t the heart to calculate the odds of that particular sequence happening to see if dealing it would take longer than the heat death of the universe but I would guess that that is not a bad comparison.

33 comments

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  1. 1
    CGM3

    And how many times in a Western film or show was the “big hand” at poker invariably won with a “royal flush” (ace high in spades)? Just once, I’d like to see the last hand go to the guy with three fours… but that wouldn’t be seen as “dramatic” by the scriptwriter, I guess.

    (Of course, there was one old movie where a cowpoke mentioned the time he was holding four kings and lost to the guy with a pair of sixes — they were pearl-handled .45s :-) )

  2. 2
    Randomfactor

    Well, the winning hand in “The Sting” was four jacks (upgraded by Gondorff from his original four threes…)

  3. 3
    Scott

    Don’t you know? Bond’s superpower is luck! He can’t fly unassisted or jump skyscrapers in a single bound, but he is a super hero just like any DC or Marvel comic hero. He has super serendipity.

  4. 4
    NitricAcid

    In one of his numerous articles compiled, I think in “On Numbers”), Asimov relates a story in which one of his fellow card-players complains that “We’ve been playing so long that the same hands are starting to show up again.” Isaac then shows the calculation that proves that this would take longer than the expected lifespan of the universe.

  5. 5
    Enkidum

    Glad to see I managed to waste your time with this!

    To be fair I’m fairly awful at calculating poker odds, and the heat death of the universe thing was just made up on the fly, but I suspect it’s the right kind of ballpark. Someone who plays a lot would probably be able to calculate the true odds in their head very quickly.

    At any rate, I very much agree with CGM3 – why can’t films with poker in them actually show people playing poker, instead of just a sequence of hands that look impressive? (Then again I guess that’s like asking why, I don’t know, scenes with gunfire have to have people being hurled backwards by single bullets, which is just as ridiculous.)

    Rounders is one very rare film where at least the poker is played reasonably straight, there’s a pretty good analysis of the final hand here: http://www.avclub.com/articles/rounders,68858/.

  6. 6
    jon1

    The universe is huge and old, and rare things happen all the time – including life.
    Lawrence Krauss

    IMO, almost *EVERY* movie (especially in the bond/action/bullets/explosions range of movies) either concludes, is based on or completely consists of highly improbable stuff. Singling out a poker game in a Bond movie to me is like pointing out that “that one christian over there believes in god.”

  7. 7
    Jenora Feuer

    Heh. I kind of liked the bridge game in the book Moonraker, myself. Drax had a hand with twelve court cards (AKQJ spades, AKQJ hearts, AK diamonds, KJ9 clubs). Bond managed to win with a 7 clubs bid, doubled and redoubled. For the non-bridge players, ’7 clubs’ means winning all thirteen tricks, despite Drax having 12 of the 16 highest cards.

    Of course, in that case, there was nothing random about the bridge game: Bond had pocketed a pre-arranged deck (actually, two pre-arranged decks, one for each of the house colours) in order to teach Drax a lesson about cheating, and had done so with full approval of the house. Drax had been cheating by using a reflective cigarette case to watch the cards he was dealing to the other players; M had asked Bond to figure out how Drax did it, and help teach Drax a lesson because he knew Bond was a card shark.

    In the original books, Bond was no stranger at all to cheating at cards.

  8. 8
    Mano Singham

    I remembered that game (I used to be a bridge player too) but had forgotten which book it appeared in. Thanks for reminding me.

    If I remember correctly, Bond pretended to be drunk during the bidding sequence so that Drax did not suspect that he was being taken for a ride.

  9. 9
    Enkidum

    @jon1: well, in the case of poker the odds are actually precisely defined. You’re right that it’s no more ridiculous than space lasers or whatever, but it’s still pretty ridiculous.

  10. 10
    slc1

    Actually, in the novels, James always played chemin de fer, not poker. I haven’t read the novels in several years but I don’t recall him ever playing poker.

  11. 11
    Sean Sherman

    There’s nothing unreal about this hand, actually. I would call it ho-hum as far odds go. Quads over quads and quads beaten by straight flushes is more rare. Though its not as uncommon as you’d expect. Google ‘Bad Beat Jackpot’ – casinos pay out a jackpot when this happens. Depending on the requirements and casino volume, it can take from one month to several to happen. I would be surprised if all given sequences of these strange possibilities haven’t already happened.

    The general version of this hysteria explains both why martingaling doesn’t work, and yet is such a popular idea. What is really unrealistic to me is the splashing of chips over other player’s chips, so nobody knows whose is whose afterwards.

  12. 12
    Doug Little

    My favorite poker hand is the dead man’s hand. I have also pulled a royal flush playing Texas no limit holdem, unfortunately it was at work over lunch when playing for quarters.

  13. 13
    Rob Grigjanis

    According to this, the odds are 1 in 158,551,976. Not that high! If there are 10,000 games played every day, it would only take about 44 years to play that number of games.

    Does Enkidum know something about the heat death of the universe that he/she isn’t telling us?

  14. 14
    Physicalist

    This hand was pretty unbelievable. Aces vs. Kings vs. Queens — and the Aces came in las

  15. 15
    Physicalist

    t (if you count the folded winning hand . . . )

  16. 16
    Doug Little

    I found the odds for specific hands in Texas holdem to be

    0.197% for a Flush
    0.144% for a Full House
    0.00139% for a Straight Flush

    Assuming simple probability just multiplying these together gives you about

    5.678×10^-12% odds for getting those 4 hands in one particular hand of Texas no limit holdem.

  17. 17
    Timothy

    “At any rate, I very much agree with CGM3 – why can’t films with poker in them actually show people playing poker, instead of just a sequence of hands that look impressive?”

    Because there are very few things in existence than watching people play poker. (And I include professional golf in that statement.)

    I loved the heat death of the universe thing! Apparently, though, it’s a real thing:

    http://en.wikipedia.org/wiki/Heat_death_of_the_universe

  18. 18
    iainr

    And 10,000 games every day is underestimating the amount of poker in the world by several orders of magnitude.

    Picking the first low level table on Pokerstars from the lobby it’s currently dealing hand number 97,601,097,166 – and that’s just one on-line poker site, the biggest but still.

    The hand number went up another 50,000 while I wrote the second half of the preceding sentence.

    It got to 97,601,232,795 while I wrote the one before this.

  19. 19
    Rob Grigjanis

    It just struck me that the calculation on the site I linked neglected to factor in that the second full house has to beat the first full house. Still, with Iainr’s observation, it probably happens regularly. Not sure what the odds would be if you restrict the scenario to licensed-to-kill MI6 agents playing evil geniuses. Maybe we’re back in heat death territory.

  20. 20
    Enkidum

    You skeptics with your so-called “facts” and “mathematics” and “real world” are spoiling my fun!

    So there may have been a tiny bit of exaggeration in my statement. Still, with an appropriate margin of error, 12 billion years (or however long it will be until the heat death of the universe) is indistinguishable from 1 year (or however long it would take to get that hand, given iainr’s points). Highly accurate, not terribly precise.

    I think we can all agree that no-limit holdem is a stupid way to take down evil geniuses, at any rate.

  21. 21
    ollie

    This sort of reminds me of the final hand in the Cincinnati Kid If I were the Kid, I would have bet the way that he did.

  22. 22
    AJS

    Most people aren’t poker players, so only recognise the highest scoring hands — runs and flushes. Even the humble prial, which will usually be worth something even without a pair to make it into a full house,

    Introducing wild cards (usually only found in draw poker games) creates new hands such as five-of-a-kind. In some schools, 5555W (five fives) beats AAAAW (five aces). In the event of a tie, the hand with fewer wild cards wins. Except that five wild cards beat anything.

  23. 23
    iainr

    No Limit Hold-em is a terrible way to take down evil geniuses.

    Error bars that hide the difference between 1 year and 12,000,000,000 indicate a terrible way of measuring time.

    Still, what’s 10 orders of magnitude between friends, eh?

  24. 24
    Enkidum

    I hereby rescind all comments I have ever made about poker, probabilities, and/or the heat death of the universe. And pretty much everything else, just to be on the safe side.

  25. 25
    Marcus Ranum

    Speaking of crazy card hands, how about a Mississippi Flush?

    Ray Wylie Hubbard
    skip to 50:16

  26. 26
    Andrew G.

    That calculation is wrong – it’s assuming independence of the four hands, but in hold’em you can’t ever assume that since so many of the cards are shared.

    I took a different approach, by running a simulation with the following parameters:

    4 players, all stay in to the end regardless of cards (not realistic, but hey)

    I define a success as: first player gets at least a flush, second player beats them with at least a full house, third player beats the second, fourth player beats the third with a straight flush.

    After five hundred million trials, I have 8 successes, so my estimate of the odds is on the order of 1 in 62.5 million, quite a bit better than the linked calculation.

    The real odds of it actually happening are rather worse of course because in many of the cases, real players would not all have stayed in all the way through.

    Probably the most impressive of the success cases was:

    player 1: 9heart 8club
    player 2: 2spade 2diamond
    player 3: Kclub Kdiamond
    player 4: Aheart 10heart
    flop: Kspade Jheart Kheart (player 3 makes his quad)
    turn: Qheart (player 4 makes his royal flush)
    river: 2heart (player 1 makes his flush, player 2 makes twos full of kings)

    (If you relax the conditions on the hands, just requiring player 1 to have at least a flush and for each player to beat the one before, you get odds more like 1 in 50 thousand, thanks to relatively common cases where there’s a flush (or fours) on the table and each of the four players has a progressively higher kicker.)

    (And yes, I was using a random number generator with enough state and entropy to generate all possible shuffled decks – most don’t)

  27. 27
    Francisco Bacopa

    I have ho idea why they call this game Texas Hold ‘Em. We always used to call it “Community” when I was growing up.

  28. 28
    chrislawson

    I don’t know if Shane was the first film to have a man blown backwards by a single bullet — made more unrealistic by the fact that the bullet came from a handgun. But there’s an interesting story to that shot. The director, George Stevens, had served in the US Signals Corps as a film reporter and had seen combat firsthand, including D-Day. Stevens wanted audiences to see what it really meant for a man to die instead of the traditional Western where the shot character would quietly crumple into a neat corpse. Of course, it’s possible that Stevens did not realise that some of the soldiers he saw die at Normandy were shot by entrenched machine guns or other artillery where the force really could knock a man backwards, or it’s possible that he his memory had made normal rifle shots more powerful than they really were, or maybe he knew full well that a handgun couldn’t do that but still wanted the audience to really feel the shot and fudged the facts for emotional clout.

    Either way, I have a great deal of sympathy with Stevens’ goal even with the factual inaccuracy (and if you know the moment from the movie, you know how it was portrayed as a terrible a waste of a human life). Certainly more so than movies that seem to think the important thing is for guns to look cool. (Worst offender: the Schwarzenegger vehicle Eraser, in which the hero holds an experimental weapon in his hands that fires a round so powerful it knocks an armoured vehicle off its wheels — a force so strong that the recoil should have smashed Schwarzenegger to a pulp.

  29. 29
    Mano Singham

    I love Shane and have seen it several times but had not noticed that effect. I agree that the film tried to show the negative side of violence with the title character trying to escape from his past.

  30. 30
    Jenora Feuer

    He wasn’t just pretending, he had been obviously drinking at the table. And, I believe, had taken some Benzedrine (or something similar) before hand to help mitigate the effects of the alcohol he was going to be drinking. He warned M up front that this would make him act a bit more cocky than usual.

    Why do I remember all this about this one scene? Probably because it was just so… unlike the movie version of Bond, but still very much in character. Bond was not a nice person; he was a career sociopath, and very good at it.

  31. 31
    GregB

    The fact that there is a straight flush severely reduces the probability of this event happening
    (which is in no way approaches anything like the “heat death of the Universe”).

    For a straight flush to be possible means there is also a flush possible, and having two players with two
    cards of the same suit isn’t all that rare in Texas Hold Em. So this really reduces (to a good approximation)
    to a straight flush beating a full house, which is a standard, mundane way jackpots are won in cardrooms
    all over the US, each day.

    Something similar to the movie scene probably happens every month, perhaps more than once.

  32. 32
    Hatchetfish

    I tend to interpret some of them, especially the Bond scenes like in Moonraker, with its explicit deception, as less about probability than as two competing card cheats. Non-canon in many/most cases, but my personal preferred reading.

  33. 33
    slc1

    I am far from being an expert on hand guns but I would suspect that a bullet from a .50 Colt Desert Eagle would probably cause the victim to be blown backward. Of course, such a hand gun didn’t exist in the late 19th Century.

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