Video Poker Paytable Calculator
Type any Jacks or Better, Deuces Wild or Bonus paytable and get its return under perfect play, as an exact fraction and a decimal, with the variance and how often every hand comes up. It solves all 2,598,960 deals in your browser in a second or two, even on a phone. Free, no account, no download.
9/6 Jacks or Better
99.543904%
- Exact return per coin
- 1653526326983 / 1661102543100
- To twelve places
- 0.995439043695
- House edge
- 0.456096%
- Variance per hand
- 19.514676
- Standard deviation
- 4.4175
- Hit frequency
- 45.4565% (1 in 2.20)
Variance as an exact fraction
53846098447064372932173011 / 2759261658693287357610000
| Hand | Pays | Combinations | Probability | 1 in | Return |
|---|---|---|---|---|---|
| Royal Flush | 800 | 41,126,022 | 0.002476% | 40,391 | 1.9807% |
| Straight Flush | 50 | 181,573,608 | 0.010931% | 9,148 | 0.5465% |
| Four of a Kind | 25 | 3,924,430,647 | 0.236255% | 423.3 | 5.9064% |
| Full House | 9 | 19,122,956,883 | 1.151221% | 86.9 | 10.3610% |
| Flush | 6 | 18,296,232,180 | 1.101451% | 90.8 | 6.6087% |
| Straight | 4 | 18,653,130,482 | 1.122937% | 89.1 | 4.4917% |
| Three of a Kind | 3 | 123,666,922,527 | 7.444870% | 13.4 | 22.3346% |
| Two Pair | 2 | 214,745,513,679 | 12.927890% | 7.74 | 25.8558% |
| Jacks or Better | 1 | 356,447,740,914 | 21.458503% | 4.66 | 21.4585% |
| Nothing | 0 | 906,022,916,158 | 54.543467% | 1.83 | 0.0000% |
How it is solved
A paytable's return under perfect play is the average, over all 2,598,960 deals, of the best of the 32 ways to hold each one. Done directly that is 2.6 million deals times 32 holds times up to 1.5 million draws, far too much for a phone. The calculator gets the same answer a shorter way.
First it sorts every five-card hand into its category once, and for every set of up to four cards counts how many hands containing that set land in each category. A hold's outcomes are then exactly the hands that contain the held cards and none of the discards. Inclusion and exclusion over the discards turns the counts already made into the counts for every hold: 80 additions per deal instead of millions of evaluations. Deals that differ only by suit are solved once and counted as many times as they occur, which leaves 134,459 to visit.
Every comparison between holds is between whole numbers. Each hold's value is put over one common denominator (the least common multiple of the numbers of possible draws, 7,669,695), so two holds tie only when they are exactly equal and a difference of a billionth of a coin still decides the play. The totals are kept as exact integers to the end, which is why the return comes out as a fraction.
How to check this
Stewart N. Ethier works two paytables completely in The Doctrine of Chances (Springer, 2010), chapter 17, and the calculator reproduces both exactly:
- 9/6 Jacks or Better (800-50-25-9-6-4-3-2-1): return 1653526326983 / 1661102543100 = 99.5439%, variance 19.514676, and 41,126,022 royal flushes out of 1,661,102,543,100 under the minimum-variance strategy.
- Full-pay Deuces Wild (800-200-25-15-9-5-3-2-2-1): return 32187682693 / 31944279675 = 100.7620%, variance 25.834618 under the minimum-variance strategy.
It also reproduces every one of the 93 paytables this site publishes, combination for combination. Pick any of them from the list above and compare with its own analysis page.
Reading the numbers
- Return: what the machine pays back per coin bet, on average, under perfect play. 99.54% means a long-run loss of 0.46 coins per 100 bet.
- Variance: the spread of one hand's result, in coins squared. Two paytables with the same return can differ a lot here, and the one that pays more for rare hands swings harder. It is the figure that sizes a bankroll.
- Hit frequency: how often a hand pays anything, even a push. It says how a machine feels to play, not what it returns.
- Return column: how much of the total each hand supplies. On 9/6 about 2% comes from the royal alone, which is why a session without one runs well below the headline figure.
Frequently asked questions
- What does the calculator assume about how I play?
- Perfect play at max coins: on every one of the 2,598,960 possible deals it holds the cards with the highest expected value for the paytable you typed, and pays the royal at the max-coin rate. That is the return printed on every published paytable analysis, and the most any strategy can get. A real player who makes mistakes gets less, and one who bets fewer than five coins gets much less, because the royal pays 250 a coin instead of 800.
- Why give the return as a fraction?
- Because it is one. Every hold's value is a whole number of winning draws over a whole number of possible draws, so the return of a paytable is an exact ratio of two integers. 9/6 Jacks or Better is 1653526326983 / 1661102543100, which is the figure Stewart Ethier gives in The Doctrine of Chances. A fraction can be checked digit for digit; a percentage rounded to two places cannot tell two close paytables apart.
- What is the tie-break setting?
- Some deals have two holds whose expected values are exactly equal, not just close. Either one is optimal and the return is the same, but they make different hands, so the frequencies and the variance depend on which one you keep. "Keep more cards" is what Ethier calls the minimum-variance strategy and is what every paytable page on this site uses. "Keep fewer cards" gives the maximum-variance one. For full-pay Deuces Wild the variance is 25.834618 one way and 25.837916 the other.
- Can I enter a paytable for Double Double Bonus or Bonus Poker?
- Yes. Pick the game first, because the Bonus games pay four of a kind by rank and so have their own rows: four aces, four 2s to 4s and four 5s to kings, and in Double Double Bonus four aces with a 2, 3 or 4 beside them and four 2s to 4s with an ace to 4. Each opens on the schedule players mean by its name (8/5 Bonus Poker, 10/7 Double Bonus, 9/6 Double Double Bonus), and every published schedule of each is in the list of known paytables.
- Why are payouts per coin?
- Because that is how returns are quoted. Type the five-coin column divided by five: a royal that pays 4,000 for five coins is 800 here. Decimals are allowed, up to three places, for schedules such as a straight flush that pays 239 for five coins.
- Is anything sent to a server?
- No. The whole solve runs in your browser, in a background thread, and no server computes anything. Free, with no account and no download.