What are the chances that five cards could permit a straight in Texas Hold Em?

Factual Questions
1

I like poker, something I’ve mentioned before.

I have noticed that when playing Hold 'Em, once five cards are on the board, it is usually possible for one of the players in the hand to hold a straight. For instance, if the board includes these cards:

ACE QUEEN TEN EIGHT DEUCE

It is possible that a player could hold a king and a jack, which would give them a straight of A-K-Q-J-10.

If the board is A-Q-J-4-3 two straights are possible; a player holding either king-ten or five-deuce would have a straight. A-10-8-7-2 would also allow two straights; either jack-nine or six-nine would make straights.

However, a board of K-Q-8-7-3 has no possible straight. Neither does A-A-A-10-6, or K-K-8-4-3.

(For anyone good at math but not familiar with the rules, for the purpose of straights, an Ace can be either high, above the King, or low, below the deuce. But not both in the same hand. A-K-Q-J-10 and A-2-3-4-5 are straights. Q-K-A-2-3 isn’t.)

Now, like I said, it is my subjective impression most five card boards have a theoretically possible straight - but what percentage of all boards would that be?

2

1940480/2598960 = 24256/32487 ~= 74.66%

Brute force Python script…

from itertools import combinations as combos

straights = [set([i, i+1, i+2, i+3, i+4]) for i in range(1, 10)]
straights.append(set([10, 11, 12, 13, 1]))
deck = [i%13 + 1 for i in range(52)]
boards = 0
successes = 0
for board in combos(deck, 5):
    boards += 1
    for straight in straights:
        if len(set(board) & straight) >= 3:
            successes += 1
            break
    
print(successes, boards)
3

There’s also a possible straight to the queen here: 89TJQ.

4

This isn’t far off what I would have guessed. Thank you!

5

I think of that as “no wraparound.”

6

This is usually called “aces swing” The latter case would be “everything swings” were it a straight.