Solved – What are the odds of drawing 7 cards that end up sequential – from a 52 card deck? (7 card poker straight)

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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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Best Answer

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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