In poker, what are the odds of drawing to an inside straight flush. Over the weekend a friend of mine actually pulled this off, and I was wondering just how incredibly lucky they were.
With a 52 card deck, there are 2,598,960 possibilities of different hands of 5 cards (there’s a special function for this on many calculators). Every suit has 13 cards and has nine possible straight flushes (A2345, 23456, etc, up to 9.10.JQK - ten if you count the royal flush as a straight flush), so there are 36 (or 40) different straight flushes. This makes your odds of getting a SF straightaway 36/2,598,960, about 1.4 thousandth of a per cent.
That said, methinks there are 200 different four straight flushes possible (each one of the 40 SFs with another one of the five cards missing each time, so 40*5), so chances of getting an incomplete straight flush are 200/2,598,960 = 7.7 thousandth of a per cent. Multiply this by 1/47 (because you have to get the one correct card out of the 47 still in the deck), and it would be 0.16 thousandth of a per cent probability of drawing the missing link.
But wait, that would mean chances to draw to it are lower than getting it straightaway? Humm. Forget what I’ve written, sorry.
Professional mathematicians required.
My math (52 x 51 x 50 x 49 x 48) says there are 311,875,200 possible hands. One of us has to be wrong…
To (I hope) make this just slightly more comprehensible the ratio for 36/2,598,960 is identical to 1/72,193.333…etc. If you’re playing five-card draw and trying to fill in an inside straight flush by pulling one card your odds are about 1/47 — 52 cards minus the five you already have in your hand.
It’s me - I realized my mistake. Ignore me (this time, anyway)
There would be 311,875,200 possibilities if you count the order of cards within your hand (e.g… K of Spades and four aces would be a different hand than four aces, K of Spades, and Ace of hearts, K of Spades, three remaining aces would be a third one). But that’s not the case.
Oops. A handful of new posts slipped in while I was typing away. You may want to forget that last post. Math is not my forte.
Can there be something more embarassing than correcting something that’s already known to be wrong?
Sorry, Running with Scissors, ignore my post as well and we’re even.
Math is not exactly my forte either but I’ll ante up.
The possible hands in a 52 card deck are 2,598,560, as stated above. The number of possible straight flushes is 40 (counting royal flushes), also as stated above. 4 suits times 10 possible runs in each suit. I checked online at various poker sites to verify this.
An inside straight is defined as one of the 3 inside cards are missing, i.e. if your inside straight flush has 2C low and 6C high, then 3C, 4C, or 5C is missing. It follows then that the possible number of inside straight flushes is 4 suits times 10 possible runs times 3 possible missing cards, or 120 inside straights. Divide that by the total number of possible hands and you get a 1 in 21,658 chance of being dealt an inside straight. Compare this with a 1 in 64,974 chance of being dealt a straight flush. The poker sites I visited stated that there is a 1 in 46 chance of completing the inside straight flush, which seems right to me.
Of course, this is assuming you are the only person being dealt from the deck. Add in three friends also being dealt cards and the odds change tremendously.
Anyone care to raise?
Grrrr. That’s “…a 1 in 21,658 chance of being dealt an inside straight flush.” But you probably knew what I meant. 
Your error was in figuring there were only 5 times as many ways to get 4 out of the 5 cards in an SF. You’re kind of right that there are 5 different ways to get the 4 cards, but you still have a fifth card - any one of 47 cards (not one of your 4, and not the fifth, and don’t talk to me about jokers right now).
There are 47540 different ways to get a ‘4 out of 5 of a SF,’ or
9400 of them.
Next, however, you have to deal with the fact that sometimes you’re filling ‘the middle,’ and sometimes ‘the end.’
For the SF’s 2-6 and 9-K:
3/5 of the time you’re filling inside, having a 1/47 chance of doing it.
2/5 of the time you can fill either end, having a 2/47 chance.
For SF’s A-5 and 10-A:
4/5 of the time you need exactly one card, so 1/47.
1/5 of the time you could take either.
So for 8 out of the 10 major cases, or 7520 of the cases:
-3/5 of the time, or 4512 of the hands, you need 1 of 47 cards = 96 successful ways to do it.
2/5 of the the time, or 3008 of the hands, you need one of 47 = 128 more ways.
For 2 out of 10 major cases, or 1880:
4/5 of the time, 1504 hands, you need 1 card of 47 = 32 ways.
1/5 of the time, 376 hands, you have 2 cards to do it = 8 ways.
96+128+32+8 = 264 total ways you can get 4 cards and then fill a SF.
Out of 2,598,560 hands, as you said.
Odds of drawing to an inside straight flush: When you draw to an inside straight flush, there is only one card in the deck that can complete the hand. A deck has 52 cards; 5 of them are already accounted for (the five you were dealt), leaving 47. So the probability of completing that straight flush is 1/47. Expressed as odds, that is 1:46.
That’s in draw poker. If it’s stud poker, or holdem, or some other variant, it becomes more complicated. Also, if there are wild cards it’s a lot easier to get that straight flush.
The total number of poker hands is ( as stated frequently above) 52C5 = 2,598,960. The tricky part is to calculate the number of hands which are “inside straight flushes”. Not being a poker player, I shall have to extrapolate from the above and assume that an inside straight flush is a holding such as 3467A, where only one card will complete the hand.
How many such hands are there in a suit? There are 10 possible straight flushes ( as noted above) and for each one there are 141 ways to convert it into an inside straight flush ( omit any of the 3 middle cards of the sequence and replace it with one of the 47 neutral cards). The total number of such hands is then 410141 =5640. The probability is thus 5640/2598960. You should expect to get one such hand out of every 461 dealt.
If you are asking the trivial question: having been dealt such a hand what is the probability that you then convert it into a straight flush, then Mandrake’s answer of 1/47 is correct
Manduck is taking the proper approach here by assuming that you already have four of the five cards you need and are merely replacing one card in an attempt to collect (“draw”) the fifth.
www.casinoplayersclubint.com/poker/poker.html+odds+of+drawing+to+an+inside+flush&hl=en&ie=UTF-8 has some good information. They say among other things
Chances of completion:
When drawing one card to:
Four cards of a Flush 1 in 4.5
Straight open at both ends 1 in 5
Straight open at one end 1 in 11
Straight open on the inside 1 in 11
Straight Flush open at both ends 1 in 23
Straight Flush open at one end 1 in 46
Straight Flush open on the inside 1 in 46
Reading the OP, it asked for the odds of drawing to an inside straight flush.
In my playing experience, the odds are that 50% of amateur players will draw to an inside straight flush. That is why they stay amateurs - and why other people make money off them. A friend of mine paid his way through college by playing “friendly” games against idiots like that.
The discussion above concerns the odds of SUCCEEDING if you are stupid enough to draw to a an inside straight flush.
I support Manduck on this; assuming you already have four cards in a straight flush, getting the fifth is 47/52.
However, I disagree with Balor. Drawing to an inside straight may be useless, but drawing to an inside straight flush can make perfect sense. Consider how many cards can improve the player’s hand:
The 12 cards that would pair up with a card already in player’s hand.
The 8 cards that would complete his flush.
The 3 cards that would complete his straight.
The 1 card that would complete his straight flush.
Even if you ignore the first 12, since a single pair is unlikely to win, there are 8+3+1= 12 cards out of 47 that can give the player at least a straight, and a probable win. 12/47 is nearly one in four.
This table cites the odds of completing an inside straight flush as 46 to 1, and getting a straight or better as a tempting 3 to 1. It’s definitely worth a draw.