# Video Poker - Jacks or Better

I have a number of questions. The first is about the banners. I'd like to know how we could optimize your income by clicking on the banners. Do you get paid on the basis of hits, unique hits or a flat fee or some other formula? Currently I only click on a banner if it's something new, but I'd be happy to click on a few banners every time I visit, if that would help generate income.

The next questions I have are about your perfect strategy for Jacks or Better table, and the practice Jacks or Better game. I hate to seem dense, but that's never stopped me from asking questions before. Where on the table do you find the rank of a hand with 1 high card (A,K,Q,J) and no penalty cards? My second question is somewhat related to the first. According to the practice game, the optimum play for a hand with unsuited A,Q,K and no penalty cards is to hold the K,Q and discard the A with the low cards. Intuitively, I would have thought that keeping the Ace is the better play. What is the advantage of dropping the Ace, and how would I determine the optimum play on this hand from the table?

You can see from my 9-6 jacks or better strategy, that there is a single line for "one high card." That is because an individual jack to ace is worth about the same.

St Flush-250

4 Aces-750

4/2,3,4-450

FourKind-250

Fullhouse-40

Flush-25

Straight-20

3Kind-10

2Pair-10

JacksBetter-5

Secondly, I am wondering which, if any, online casinos currently advise the player of a shuffle in blackjack (multi-deck, of course). Also, do you know, among the majority who do not, which shuffle after each hand and which just do not advise of a shuffle (although it actually occurs after many hands)? It would be great to have this knowledge. A follow up question would be, if they do indeed shuffle at regular casino intervals, can a player assume that if he enters a private table that he beings with a full shoe? Thanks again for your great web site, and I look forward to your response to my questions.

So, you're much better off playing five quarters in 9-6 Jacks or Better.

RF:800There is a bank of these at Harrah's in East Chicago Indiana, on a progressive jackpot. Any info would be appreciated.

SF:50

4Aces:160

4kind(2,3,4):80

4kind(5-K):50

FH:7

flush:5

straight:4

trips:3

2PR:1

J’s or better:1

^{5}=32 ways to play the hand. I analyze each way and take the play with the greatest expected value. To determine the expected value of a play you have to analyze all the ways the replacement cards can fall and score each hand. In the case of throwing all five cards away there are combin(47,5)= 1533939 possible replacement hands. The total number of hands that must be analyzed to determine the best play of a specific hand is combin(47,5)+5*combin(47,4)+10*combin(47,3)+10*combin(47,2)+5*47+1, which coincidentally also equals 2598960. So if we took no short cuts at all we would have to analyze 2598960

^{2}= 6,754,593,081,600 hands. Just reducing the initial hands to 191659 we still have 498,114,074,640 hands to analyze. Clearly more short cuts are in order. It would take a desktop computer several hours at least to work through this many hands. Personally I don’t actually score any hand but use carefully chosen formulas to determine the probability of improving a hand. For example with any pair and 3 singletons the probability of improving the hand to a two pair is always the same. Things get more complicated with straights and flushes but still manageable. My program can calculate the expected return for a game of jacks or better in about one minute. Considering it used to take me over a day I’m rather proud of it. I hope this answers your question.

^{1/2}*4.416 = 44.16. You can then use this information to determine what is the probability of winning or losing more than a specified number of units. For example the winning or losing within two standard deviations, or 88.31 units, is 4.55%. There is more to it than this. Please see my part on the standard deviation in my section on the house edge.

^{1/2}*5*4.417542 = 22.08771. The standard deviation of 4 bets of $2.50 in 4-play jacks or better is 4

^{1/2}*$2.50*5.041215 = 25.20608. So you are better off betting the smaller total amount in 1-play. Interestingly you can double the total amount bet in 4-play and the standard deviation only goes up by 14.12%.

1=0.97434 + j*(0.024686/940)

j = (1-0.97434)/(0.024686/940) = 977.33182.

So the breakeven point is a meter of 977.33 bet units or $4886.66. This assumes perfect 940/9/6 strategy. However few people know 940/9/6 strategy. If using 800/9/6 strategy then we would use the 800/9/6 table:

1 = (0.99543904-0.01980661) + j*(0.01980661/800)

j = (1 - (0.99543904-0.01980661))/(0.01980661/800)

j = 984.2197

So if using 800/9/6 strategy the jackpot would need to reach 984.22 bet units or $4921.10.

^{0.5}= 610.02. The standard deviation of 10,000 hands of 1-play is 195.149

^{0.5}= 441.75. As long as the total final hands are the same, 10-play will always be 38.1% more volatile, in 9/6 Jacks or Better. For more information visit my section on the standard deviation in n-play video poker.

### Possible Outcomes in 9/6 Jacks or Better

Hand | Combinations | Probability |

Four to royal + straight flush | 299529168 | 0.000015 |

Four to royal + flush | 7005972000 | 0.000351 |

Four to royal + straight | 3420857076 | 0.000172 |

Four to royal + pair | 10793270244 | 0.000541 |

Four to royal (non-paying) | 21844510692 | 0.001096 |

Royal flush | 490952388 | 0.000025 |

All other | 19889375425632 | 0.9978 |

Total | 19933230517200 | 1 |

The expected number of royals for 170 four to a royals is 170/88.33 = 1.92. The probability of seeing zero with a mean of 1.92 is e^{-1.92} = 14.59%.

^{-0.1}= 90.48%. The reason it is not 90% is because sometimes you will get more than one royal per session. The expected number of royals in 50 sessions is 0.1 × 50 = 5. The probability of zero royals in 50 sessions can be closely approximated at e

^{-5}= 0.67%. The exact probability is (39,999/40,000)^(200,000) = 0.67%, as well.

^{-3.9616}= 1.9%.

The math gets messier with mutli-line games. I think the easiest way to answer the question is by random simulation. My video poker appendix 6 shows the probability of getting at least one royal per hand in 50-play 9/6 Jacks or Better is 0.00099893. Each hand of $1 50-play costs $250. So you would have played 3,200 initial hands. The expected number of hands with a royal in 3,200 hands is 3.1966. By the same method of approximation, the probability of getting zero royals is e^{-3.1966} = 4.09%. The exact answer, based on the simulation results, is (1-0.00099893)^3200 = 0.04083732, or 4.08%.

### Covariance in 9/6 Jacks or Better Spin Poker

Lines | Covariance |

2 | 1.99 |

3 | 3.70 |

4 | 9.62 |

5 | 15.27 |

6 | 19.53 |

7 | 23.37 |

8 | 27.94 |

9 | 33.46 |

Let’s look at an example of 9-line 9/6 Jacks or Better. The variance of the base game is 19.52. The covariance is 33.46. So the total variance is 19.52 + 33.46 = 52.98. The standard deviation is 52.98^{1/2} = 7.28.

Update: Since this answer was published, the Wynn removed all their 9-6 Jacks machines except at the $5 denomination and higher.

Loosely stated, skewness is going to correlate with how often you get a win in a session. In Jacks or Better, for the most part, you are not going to get a winning session over a few hours if you do not hit a royal. You can sit down at Double Double Bonus and be a winner after a few hours more often because of the big quad payouts. Because most people are subject to cognitive biases, the pain from a loss is twice the pleasure from a win. People do not really play Double Double Bonus because they like the variance, they play because they have a greater shot at winning. The following table shows some key statistics for four common video poker games. It is interesting to note that skew is greatest for Jacks or Better.

### Key Video Poker Statistics

Statistic | JoB — 9/6 | BP — 8/5 | DDB — 9/6 | DW — NSUD |
---|---|---|---|---|

Return | 0.995439 | 0.99166 | 0.989808 | 0.997283 |

Variance | 19.514676 | 20.904113 | 41.985037 | 25.780267 |

Skew | 147.114643 | 134.412152 | 66.495372 | 101.23991 |

(Excess) Kurtosis | 26,498 | 23,202 | 6,679 | 14,550 |

JoB — 9/6 = Full pay Jacks or Better

BP — 8/5 = Standard pay Bonus Poker

DDB — 9/6 = Standard pay Double Double Bonus Poker

DW — NSUD = "Not so Ugly Ducks" Deuces Wild

How can knowing this actually help the video poker player? I suppose one could say that a game with a large skew has a greater chance of a loss over a session of a few hours. For example, in Jacks or Better, if you don’t hit any royals, the house edge will probably eventually grind your bankroll down. However in a game like Deuces Wild or Double Double Bonus, the second highest wins can pull you out of the hole over a session. In other words, the skew keeps you from winning when you are not hitting royals. Knowing the skew won’t increase your odds, but it is mentally helpful to know what to expect. So, the next time you take a beating in 9/6 Jacks, blame it on the skew.

My thanks to Jeff B. for his help with this question.

### 9/6 Jacks or Better Return Table with Optimal 9/6 Strategy

Hand | Pays | Combinations | Probability | Return |

Royal Flush | 800 | 493512264 | 0.000025 | 0.019807 |

Straight Flush | 50 | 2178883296 | 0.000109 | 0.005465 |

Four A | 25 | 3900253596 | 0.000196 | 0.004892 |

Four 2-4 | 25 | 10509511320 | 0.000527 | 0.013181 |

Four 5-K | 25 | 32683402848 | 0.00164 | 0.040991 |

Full House | 9 | 229475482596 | 0.011512 | 0.10361 |

Flush | 6 | 219554786160 | 0.011015 | 0.066087 |

Straight | 4 | 223837565784 | 0.011229 | 0.044917 |

Three of a kind | 3 | 1484003070324 | 0.074449 | 0.223346 |

Two pair | 2 | 2576946164148 | 0.129279 | 0.258558 |

Pair | 1 | 4277372890968 | 0.214585 | 0.214585 |

Nonpaying hand | 0 | 10872274993896 | 0.545435 | 0 |

Total | 19933230517200 | 1 | 0.995439 |

Using the probabilities above, but applying them to the 10/7 Double Bonus pay table, we get the following return table.

### 10/7 Double Bonus Return Table with 9/6 Strategy

Hand | Pays | Combinations | Probability | Return |

Royal Flush | 800 | 493512264 | 0.000025 | 0.019807 |

Straight Flush | 50 | 2178883296 | 0.000109 | 0.005465 |

Four A | 160 | 3900253596 | 0.000196 | 0.031307 |

Four 2-4 | 80 | 10509511320 | 0.000527 | 0.042179 |

Four 5-K | 50 | 32683402848 | 0.00164 | 0.081982 |

Full House | 10 | 229475482596 | 0.011512 | 0.115122 |

Flush | 7 | 219554786160 | 0.011015 | 0.077102 |

Straight | 5 | 223837565784 | 0.011229 | 0.056147 |

Three of a kind | 3 | 1484003070324 | 0.074449 | 0.223346 |

Two pair | 1 | 2576946164148 | 0.129279 | 0.129279 |

Pair | 1 | 4277372890968 | 0.214585 | 0.214585 |

Nonpaying hand | 0 | 10872274993896 | 0.545435 | 0 |

Total | 19933230517200 | 1 | 0.99632 |

You can see the return is 99.63% playing 9/6 strategy on a 10/7 machine. You gain 0.63% from the better pay table but lose 0.54% from errors, for a net gain of 0.09%.

### 9/6 Jacks Royal Combinations

Cards Held | Combinations | Probability |
---|---|---|

0 | 1,426,800 | 0.002891 |

1 | 16,805,604 | 0.034053 |

2 | 96,804,180 | 0.196154 |

3 | 195,055,740 | 0.395240 |

4 | 152,741,160 | 0.309498 |

5 | 30,678,780 | 0.062164 |

Total | 493,512,264 | 1.000000 |

### Combinations on theDraw in Video Poker

Discards | Combinations |

0 | 1 |

1 | 47 |

2 | 1,081 |

3 | 16,215 |

4 | 178,365 |

5 | 1,533,939 |

The least common multiple of all those combinations is 5×combin(47,5)= 7,669,695. Regardless of how many cards the player discards, the return combinations should be weighted so that the total comes to 7,669,695. For example, if the player discards 3, there are 16,215 possible combinations on the draw, and each one of them should be weighted by 7,669,695/16,215 = 473.

So the total number of combinations in video poker is 2,598,960 × 7,669,695 = 19,933,230,517,200 . For more on how to program video poker returns yourself, please see my page on Methodology for Video Poker analysis.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.