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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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    $\begingroup$ 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. $\endgroup$ – Hao Ye Jun 25 '14 at 5:01
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    $\begingroup$ 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' experiment). $\endgroup$ – Glen_b Jun 25 '14 at 5:23
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    $\begingroup$ 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? $\endgroup$ – whuber Jun 25 '14 at 22:23
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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 $\approx 0.000979$.

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  • $\begingroup$ 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) $\endgroup$ – The Scrum Meister Jun 25 '14 at 5:10
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    $\begingroup$ 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.) $\endgroup$ – Hao Ye Jun 25 '14 at 5:26
  • $\begingroup$ 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. $\endgroup$ – Hao Ye Jun 25 '14 at 5:45
  • $\begingroup$ @TheScrumMeister No, the odds are more like 1 in 5 million. Hao Ye has not accounted for the sequential nature of the straight. $\endgroup$ – 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: $1.94*10^{-7}$.

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    $\begingroup$ 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) $\endgroup$ – 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).

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