Tuesday, March 07, 2006

1 in 649,740

Last night I got dealt an ace and ten in clubs. The flop: K Q J in clubs. I got a royal flush on the flop. This site says the odds of getting a royal flush in Texas Hold 'Em are 1 in 649,740. This means that you have to play 324,870 hands before you even have a 50% chance of having it. This site reports that the odds of getting struck by lightning are 576,000 to 1 (so I better watch out), and the odds of becoming President or getting killed by a falling part from an airplane are 10,000,000 to 1, so I'm not sure if I want to get 14 more royal flushes in my life. And the odds are about the same as those for drowning in a bathtub.


  1. Heh, see this is what I mean about statistics! The odds are actually better than that.

    See in those 300,000 times you are not just interested in getting the royal flush once- you could get it twice, three times, or every single time. So while it is accurate to say that you need to play that that many times to get a 50% chance of achieving the royal flush exactly once, you need to play it less to have a 50% probablity of getting at least once.

    I would work out exactly how many times... but it'd be rather hard.

  2. ... apart from that's wrong. You actually need MORE than 300,000 times, because the situation is a little more complex. My argument is incorrect, however.

    Basically, you have a probability of achieving something of 1/n. The probablity of not doing this is 1-1/n. So the probablity of achieving our event is 1-1-1/n. Now how about achieving it once if we repeat our experiment twice- well it will be 1 minus the probailiy of not getting it twice- or 1-(1-1/n)^2.

    So to get our answer to we must solve

    1-(1-1/n)^m=1/2 where m is the amount of times we are performing the event.

    We get


    taking logs


    In this case, the answer is you need to play 450365.1026 hands to have a 50% probablity of getting at least one royal flush....

  3. Thanks. I never took statistics, because it has the same effect on me as NyQuil, but without the good feelings.

    And actually I did the math myself on the original statistic and it turns out that those are the odds of getting a royal flush on the first five cards (about 0.00015%), but in Texas Hold 'Em you get two more cards to work with--i.e. you get 7 cards to make the best 5-card hand possible. I'm not exactly sure how that changes the math.

  4. heh, not to fill this comments page with me, but I think the effect of having seven cards to get a 5 card combination basically multiplies the probability by 7C2 (7!/(5!2!) where n!=n*(n-1)*(n-2)..) As to doing the relevant calcualtions from here on... I can't be bothered. I'm actually not too sure about all this, having not done pure probability for two years, instead studying statistics.

  5. oh and incidentally....did you manage to get many chips out of the hand?

  6. I think I got as much as I could. The blinds were low at the time, and I was pretty sure they both had a straight and thought we'd split the pot. I bid 60 at the end, netting 120 from the other two guys. I think if I would have bid any more, then one of them would have folded, and if I bet 120 they both would have folded.