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How Hopeful Are Four-Card Straights and Flushes at Video Poker?

4 July 2001

Video poker aficionados know the agony and the ecstasy of starting hands comprising four-card straights, flushes, and straight flushes. Agony when the cards you need prove elusive. Ecstasy when they chunk neatly into place.

These emotions are understandable. Everybody's glad to win and sad to lose. Less comprehensible is the fact that few solid citizens realize the likelihood of success so they suspect improbity, conspiracy, perfidy, or worse, in failure.

The numbers are relatively easy to determine. I'll walk you through them for games with nothing wild and with one joker.

First, assume a double-ended outside four-card straight. Maybe 3?4-5-6 or 9-10-J-Q. With no wild cards, you have eight ways to form the straight. Any of four twos or four sevens in the first example, any of four eights or kings in the second. You've already taken five cards from the 52-card deck ?? you hold four and dumped one. So 47 remain, of which eight will do the job. Your chance of joy is 8/47, or 17.0 percent. In a game with one joker, 48 cards remain from the 53-card deck, of which nine will meet the requirement. Here the chance is 9/48, or 18.8 percent.

Instead, start with an inside or a single-ended outside four-card straight. The former might be 5-6-8-9, the latter A?2?3-4 or J?Q?K-A. If nothing's wild, only four cards out of 47 will do, and 4/47 is 8.5 percent. A single joker means that five cards out of 48 will complete the straight, and 5/48 is 10.4 percent.

Similarly for four-card flushes. You've got four of a suit and five cards have been eliminated from the draw. With nothing wild, nine of the 13 cards in the suit remain, providing a probability of success of 9/47 (19.1 percent). A joker means you're looking for any of ten cards, so your chance is 10/48 (20.8 percent).

Four-card straight flushes get a little more involved. Obviously, you'd be happiest seeing the straight flush. But you'd presumably be satisfied with the consolation prize of a flush or a straight.

Start with a double-ended outside four-card straight flush, nothing wild. Two cards are needed for the straight flush ?? 2/47 (4.2 percent). Seven cards yield a flush but not a straight flush ?? 14.9 percent. And six cards make the straight but not the straight flush ?? 6/47 (12.8 percent). In all, 15 cards will work, one way or another ?? 15/47 (31.9 percent).

A joker in the deck means that three cards will make a straight flush ?? 3/48 (6.2 percent). Seven will still give you a flush ?? 7/48 (14.6 percent). And six are good for a straight ?? 6/48 (12.5 percent). Altogether, that's 16/47 (34.0 percent).

What about an inside or single-ended outside four-card straight flush? When nothing's wild, chances are 1/47 (2.1 percent) of the straight flush, 8/47 (17.0 percent) of the flush, and 3/47 (6.4 percent) of the straight. Combined, any of 12/47 (25.5 percent) will do. A joker changes the prospects to 2/48 (4.2 percent) for a straight flush, 8/48 (16.7 percent) for a flush, and 3/48 (6.2 percent) for a straight. In all, that's 13/48 (27.1 percent).

These figures are summarized in the accompanying table.
---------------------------------------------
Probabilities (in percent) of Finishing
with a Straight, Flush, or Straight Flush

 starting hand nothing wild one joker wild str flush str-fl str flush str-fl outside str 17.0 -- -- 18.8 -- -- inside/1-sided str 8.5 -- -- 10.4 -- -- flush -- 19.1 -- -- 20.8 -- outside str-fl 12.8 14.9 4.2 12.5 14.6 6.2 inside/1-sided str-fl 6.4 17.0 2.1 6.2 16.7 4.2

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The chances are tepid, at best. And, disheartening, considering that four-card straights, flushes, and straight flushes are rare to begin with. But defeat shouldn't taste all that bitter if you don't make the hand. Because when you do, victory is so much the sweeter. As the patron poet of punters, Sumner A Ingmark, put it:
Were losing not a great ordeal,
Winning would have less appeal.

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Best of Alan Krigman
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.