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The math hitting a straight flush ?

Discussion in 'General discussions' started by Profilman, Oct 27, 2015.

  1. Profilman

    Profilman Member

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    My friend ask me when he hit a straight flush. He said what are the odds to hit a straight flush ?
    I was like.... I dont know. 8-)
    Then i was doing the math like a flush must be 25 % when there are 52 cards and 13 out there to hit but you are holding 2 cards out of them and maybe take them away, and now we have the numbers and there should be 3 cards to hit like 5 6 7 8 9 and you are holding like 8 9 and maybe take them away and divide by how many times and something :crazy: :c: :d: :heart: :s:

    Does anyone know the math ? :mrgreen:
     
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  2. impactpoint

    impactpoint Active Member

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    To figure the odds....first recognize that there are 10 ways to make the flush....1st way is help, help, help....in that way you make the flush on the flop, another is no help, help, help, help...you make the flush on the turn, another is help, no help, help, help...again you make the flush on the turn but in a different way...you will find that there are ten combos to make the flush. Then you add up the probabilities from all ten combos...example....for the first way to make it on the flop.....first card-11/50 chance for help, 2nd card 10/49 chance for help, third card-9/48 chance for help....multiply the three probabilities to get the total probability of making the flush on the flop=.84%. add up all 10 ways and you come up with 6.4%
     
  3. Aarnimetsa

    Aarnimetsa Well-Known Member
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    I'd say, it depends. On holdem, your probability of making a straight flush depends a lot on your cards. With suited and connected cards you'll have a better chance of making a straight flush.

    If we're just ignoring the hands we are dealt and think about 7 card hands. There are 133784560 different 7 card poker hands from 52 cards ( 52 choose 7 ).

    Basically as impactpoint said, we're having 10 different straight flushes that can happen (if we include Royal flushes). For a royal flush, from a certain suit, there is only one 5 card combination that will hit it. We can get how many 7 card combinations to hit a certain suit's royal flush by getting how many ways there are to select 2 cards from 47 remaining cards (47 choose 2 = 1081). When there are 4 different suits, the opportunities to hit a Royal Flush are 4324 combinations, out of 133784560 total 7 card poker hands. So getting a Royal flush on a random hand, probability for that is 0.003%.

    For any other flush than Royal flush, we're having a bit different odds. We can't select from 47 cards. Why not? One card in there can't be selected (like if we're having K-9 straight flush, we can not select A from the flush, because then we would have a Royal flush, not a K high straight flush). So for any other straight flush, we're having only 46 choose 2 combinations of other cards we can select ( 1035 combinations per suit ). So if we're only interested about non-Royal flushes, it's going to be 4140 * 9 = 37260 opportunities. When we divide it by total opportunities, it's 37260 / 133784560 , which is about 0.028%.

    If we're including both straight flushes and Royal Flushes, we're getting any straight flush in about 0.03% of our random hands. So like about 1 in 3333 hands would be a straight flush, if every hand was played to the end.

    In real life, it will depend on which games you play, how many games you'll get to the river, etc. In general, on average you're playing hands that will hit straight flushes better than a random hand, but for something like a 6-max player playing about 25% of hands, it will probably still require about 6-8 times as many played hands as naked odds to hit a royal flush to actually see one, as it will be so rare to get to the river with one (20000-30000 hands). For a full ring player it will take even more hands and a headsup player will see more straight flushes per a certain amount of hands.
     
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  4. Profilman

    Profilman Member

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    Thank you Aarnimetsa for your answer.

    133.784560 different cards from 7 cards that you can create out of 52 ? wow didn't think about that :crazy:

    Where do you get 1081 from ?
     
  5. Aarnimetsa

    Aarnimetsa Well-Known Member
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    How many different 2 card combinations you can make from 47 cards. You can get it quickly from google search by just searching "47 choose 2". It is a shorthand for (47 * 46) / (2 * 1).

    Pretty much the same way you can get how many different 7 card combinations you can get from 52 cards that you choose, when order does not matter => (52 * 51 * 50 * 49 * 48 * 47 * 46) / (7 * 6 * 5 * 4 * 3 * 2 * 1) => 52 choose 7 => 133784560
     
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  6. Profilman

    Profilman Member

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    Thank you. I didn't know what the word choose is in math.
     
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