How often each hand lands in Jacks or Better

Most hand-frequency tables count the five cards you are dealt. That table is fixed by arithmetic: 2,598,960 possible hands, 4 of them royal flushes. A video poker machine lets you hold some cards and replace the rest, so the hand you finish with lands far more often than the hand you start with. This post gives those finishing frequencies for 9/6 Jacks or Better, played correctly, and shows where the 99.54% return comes from.

The table after the draw

These are the standard figures for 9/6 Jacks or Better (a full house pays 9, a flush 6) with the best hold chosen every time. The royal pays 800 per coin at a five-coin bet. "Once in" is one divided by the probability, rounded. "Return" is the pay multiplied by the probability, in percentage points of the amount bet.

Check any row yourself: 25 × 0.236% = 5.9 points for four of a kind, and 3 × 7.445% = 22.3 points for three of a kind. Add the column and you reach the machine's 99.54%. The rounded rows land a hair above that, which is rounding and nothing else.

Dealt against finished

The gap between the two tables is the draw doing its work. A royal flush is dealt once in 649,740 hands (2,598,960 ÷ 4). Finished with, it comes once in about 40,400, roughly sixteen times as often, because every hand with four royal cards is a chance to complete it. A flush is dealt about once in 509 hands. Correct play finishes one about once in 91, because four to a flush is worth drawing to.

Notice how close the next three rows finish: full house 1 in 87, straight 1 in 89, flush 1 in 91. Those hands are worth 9, 4 and 6 coins, so the order of the paytable is not the order of frequency.

Where the return actually comes from

The royal flush gets the attention, but it supplies 1.98 of the 99.54 points. Look at the three most common winners instead:

• Jacks or better: 21.46 points
• Two pair: 25.86 points
• Three of a kind: 22.34 points

Together that is 69.66 points, a little over 70% of everything the machine returns, from hands that pay 1, 2 and 3 coins. Add the full house and the total is 80.02. The quads, straight flushes and royals that give the game its reputation supply just 8.44 between them.

What a typical hand looks like

Add the winning rows: 0.0025 + 0.0109 + 0.236 + 1.151 + 1.101 + 1.122 + 7.445 + 12.928 + 21.459 comes to about 45.46%. So 54.54% of hands pay nothing, and the other 45.46% pay something. Of those, the jacks-or-better row at 21.46% only returns your own coin. That leaves roughly 24.0% of hands, about one in four, that pay more than you put in.

Put another way, a hand that ends with a refund is nearly as common as a hand that ends with a profit, and both are outnumbered by the hands that end with nothing. A machine returning 99.54% is not one that pays you back most of the time. It is one that pays back small amounts often and large amounts rarely, and the long-run average works out close to even. How far a real session strays from that average is a separate matter, covered in the post on variance.

Using the table

Three things follow, all checkable against the numbers above.

1. A session of a few hundred hands is dominated by the first three winning rows. Expect a pair of jacks or better about every fifth hand and two pair about every eighth.
2. Quads, straight flushes and royals together appear in about 0.25% of hands (0.236 + 0.0109 + 0.0025), roughly once in 400. They carry 8.44 points of the return. They are worth playing for, but not worth planning around.
3. The royal flush's weight is small but not trivial. For its share and what happens when it is removed, see why the royal is worth two points of return.

None of this changes which cards to hold, which is set by comparing the expected pay of each choice (see the hold order). What it does is show why those choices matter. Correct play is the reason a flush finishes one hand in 91 instead of one in 509, and each such finish is a few extra points of return that a lazier strategy would give up. For how to read the pay column of any machine before you sit down, see reading a video poker paytable.