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While playing a friendly game of Texas Hold'em poker, a player drew a 7 card straight.

Although in texas hold'em a player may only use 5 of the possible 7 cards, the discussion about odds immediately came up. What are the odds of getting a 7 card straight?

For those unfamiliar with poker, the question can be asked this way: What are the odds of drawing 7 cards from a 52 card deck, with those cards ending up sequential? *

I tried searching the vast knowledge of the internet, but was unable to come up with an answer. the closest i get is the probability of drawing a 5 card straight in a 5 card stud game (0.00392465), but i got lost trying to add the probability of the next 2 cards - due to the complexity of the straight (the next 2 cards can complete the straight - if the first 5 cards drew 4.5.7.8.9 and the next 2 cards were 6.10).

Any help or pointers on this subject would be extremely helpful. calculating a straight

*) An Ace card can be used to start a low straight or to complete a high straight - both A.2.3.4.5.6.7 and 8.9.10.J.Q.K.A are legal. but it cannot be used to wrap - J.Q.K.A.2.3.4 is not legal.

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    You are correct for noting that you can't just piggyback off the calculation of a 5-card straight. Per your example, there are 7-card straights that are formed by adding 2 new cards to 5-card hands that are *not* straights. – Hao Ye Jun 25 '14 at 05:01
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    Beware applying calculations that apply to events specified in advance to something observed *post hoc*. That is, if you see something that may be unlikely and say "what are the chances of *that*?", the calculations done as if it were an event specified before the draw are nonsensical (as simple thought experiments can show - e.g. the '[wheelbarrow full of distinguishable dice](http://stats.stackexchange.com/questions/67721/is-there-a-law-that-says-if-you-do-enough-trials-rare-things-happen/67723#67723)' experiment). – Glen_b Jun 25 '14 at 05:23
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    To follow up on @Glen_b's point, even if this is not a *post hoc* question, there remains important ambiguity about the circumstances. Do you want to know the chance that *you in particular* draw a seven-card straight on *this hand,* or perhaps do you want the chance that *somebody playing this round* draws a seven-card straight? Or maybe the chance that *somebody in your card-playing group* will draw such a straight sometime during the course of this evening's play? Or maybe the chance that you will witness this hand sometime in your playing career? – whuber Jun 25 '14 at 22:23

3 Answers3

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In short, the probability of a 7-card straight when drawing 7 random cards from a standard deck of 52 is $0.000979$.

To calculate this value, we note that all 7-card hands are equally likely, of which there are ${52 \choose 7} = 133,784,560$ possibilities.

Next, we compute the number of 7-card straights. Ignoring suit, we note that there are $8$ possible straights (starting with {A, 2, 3, 4, 5, 6, 7} through {8, 9, 10, J, Q, K, A}). For each card in the straight, there are 4 possibilities for the suit, such that there are $4^7 = 16384$ ways to assign the suits to the 7 cards. However, $4$ of these suit assignments yield straight flushes (all clubs, all diamonds, etc.), so the actual number of suit assignments that can yield a straight (but not a straight flush) is $16384 - 4 = 16380$.

Putting all this together, there are $8 \times 16380 = 131,040$ possible 7-card straights out of $133,784,560$ possible 7-card hands, yielding a probability of $\approx 0.000979$.

Hao Ye
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  • Interesting. so the odds are about 1000:1? I guess a 7 card straight is less memorable than 4 of a kind (which according to our memories seem more common than a 7 card straight) – The Scrum Meister Jun 25 '14 at 05:10
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    Actually, in 7-card stud, a 4 of a kind *is* more likely than a 7-card straight: There are 13 possible ranks for the 4 of a kind, and the remaining 3 cards can be chosen from the any of the other 48 cards in the deck (${48 \choose 3} = 17296$) yielding $13\times17296 = 224848$ possible hands, or slightly under twice as as likely as a 7-card straight. The important thing to note is that the probability of 5-card hands in 5-card stud do not match up with the probability of 5-card hands in 7-card stud. (However, the relative probability of the hands is the same, and so the rankings are the same.) – Hao Ye Jun 25 '14 at 05:26
  • Sorry, in my previous comment, I used "relative probability", when I meant "relative ranking". The probabilities of specific hands do not maintain the same ratios with each other, though the ordering remains the same. – Hao Ye Jun 25 '14 at 05:45
  • @TheScrumMeister No, the odds are more like 1 in 5 million. Hao Ye has not accounted for the sequential nature of the straight. – Alexis Jun 25 '14 at 16:52
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Hao Ye has not quite got it. Each one of those 131,040 possible 7-card straights can be ordered $7!=5040$ ways, only one of which is the cardinal ordering from least to greatest. So the probability of observing a sequential 7-card straight is much lower: $1.94*10^{-7}$.

Alexis
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    I worded the question wrong. the 7 cards do no need to be **drawn** in sequence, rather: after drawing all 7 cards, they need to end up sequential. (that's what a poker straight is. see the example in the question) – The Scrum Meister Jun 25 '14 at 16:56
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Here is a little different approach for those of us who prefer to make the computer work hard instead of making ourselves think hard. I wrote the following R function to simulate a single draw and check if it is a straight:

straightsim <- function(ncards) {
    hand <- sample( rep(1:13, 4), ncards )
    hand <- sort(hand)
    hand2 <- hand - hand[1]
    straight <- 0
    if( all( hand2 == 0:(ncards-1) ) || all( hand2 == c(0, (14-ncards):12) ) ) {
        straight <- 1
#       print(hand)
    }
    straight
}

This ignores the suits, so does not distinguish a straight flush from a regular straight (will slightly over estimate the probabilities if you want straights that are not flushes as well).

When I ran it 1 million times (1,000,000) on my laptop it found 978 straights for an estimate of 0.000978 (surprisingly close to Hao's answer). When I ran it for 50 million (50,000,000) times on a 100 core cluster it found 48,965 straights for an estimated probability of 0.0009793 (approximate 95% confidence interval: 0.0009707 - 0.0009880).

Greg Snow
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