Video Poker - Jacks or Better

Thanks Denis for your kind words and your interest to help keep the site financially healthy. When you first wrote just clicking my banners helped. However, as I update my answer in 2013, it is acquisitions that put rice on my table. It doesn't do much good to click on the banner if you don't at least sign up for an account, and preferably make a deposit.

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.

I doubt that Casino Niagara would have the "full pay" pay table that my Java game is based on. With little competition they can be stingy and people will still play. I'm afraid I don't have any strategies available for other pay tables. I suspect Casino Niagara offers 8/5 jacks or better, which pays 8 for a full house and 5 for a flush. Assuming perfect strategy, this has a return of 97.30%. Using perfect strategy for full pay video poker, as found on my site, on this game the return would be 97.29%. The two strategies are almost the same and you are only giving up 0.01% by using my strategy on an 8/5 machine. Also, I assume maximum coins bet because that is what the player should do. If you play less than maximum coins you will only get 250 per coin on a royal flush, causing a reduction in the rate of return of 1.36%.

The return is 99.9367%.

Thanks for the kind words. Yes, the probability of a three of a kind depends on the pay table, which affects player strategy. My video poker program always makes the optimal play for every hand by looping through all the possible cards on the draw. However, creating a strategy in writing is very time consuming.

I assume by "10/7" you mean double bonus. As my video poker cheat sheet shows, the return for that game, with five coins bet, is 100.17%. However, if $5 is too rich for your blood, then you can use my video poker analyzer to get the return for a one-coin bet. Just put in 250 per coin bet for a royal flush. The calculator defaults to 4,000, so change it to 1,250. Press "analyze" and you'll see the return is 99.11%.

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

You can get close to optimal strategy for just about any video poker game by using my video poker strategy maker.

There are combin(52,5)=2598960 possible combinations of the first five cards. You don’t have to analyze all of them. Personally I break them down into 191659 different kinds and weight each one with the number of similar hands. For example the odds are the same with four aces and a king singleton regardless of the suit of the king. You don’t have to analyze four hands for each possible suit of the king, just one of them and multiply by four. Once you have a hand there are 2⁵=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 2598960²= 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.

The standard deviation of 100 hands of 9/6 Jacks or Better is 100^1/2*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.

For those who don't know Power Poker is a Microgaming term for 4-play video poker. $1 video poker has much more volatility than 25 cents 4-play. With more volatility the probability of ruin is greater, but so is the probability of a big win.

From my video poker appendix 3 we can see the standard deviation for 1-play jacks or better is 4.417542. The standard deviation for 4-play jacks or better is 5.041215. Keep in mind these figures are per hand and relative to the betting unit. Adjusting for bet size and number of hands the standard deviation of $5 bet in 1-play jacks or better is 1^1/2*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%.

There are two spreads for forming a straight with K/J (AKQJT and KQJT9) and only one spread with A/J (AKQJT).

No! Absolutely not!

Leave it to a couple math geeks to argue over the eighth decimal place. I checked the exact combinations of each hand and you are right. The exact value is 5*(84412/178365), or 2.36627140974967 to 15 decimal places. Whatever you are using obviously carries floating point arithmetic to more decimal places than the internal Java calculator.

My Jacks or Better section shows the return of a 940/9/6 game to be 0.999030. The return from the royal is 0.024686. So the return from the other hands must be 0.999030-0.024686 = 0.97434. Although the probability of a royal is shown as 0.000026, that is only to two significant digits. Let’s use the return divided by the win, or 0.024686/940 as the probability. If j is the jackpot amount solve for j in the following equation:

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.

If a royal flush paid the same as a straight flush then 9/6 Jacks or Better would have a return of 98.03%.

Yes. Let’s assume you are playing 9/6 Jacks or Better. The variance per final hand is n*1.966391 + 17.548285, where n is the number of plays. So the variance per hand in 10 play is 10*1.966391 + 17.548285 = 37.2122, and in 1-play is 1*1.966391 + 17.548285 = 19.51468. The variance of 1,000 initial or 10,000 total hands of 10-play is 10,000*37.2122 = 372,122. The variance of 10,000 hands of 1-play is 10,000*19.51468 = 195,149. However, standard deviation is what I think we should be talking about, which is the square root of the variance. The standard deviation of 10,000 hands of 10-play is 372,122^0.5 = 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.

In 9/6 Jacks or Better with perfect strategy you will see a royal on the draw once every 40,601 hands, but four to a royal once every 460 hands. For every royal you see, you will be one card away 88.33 times. Of the four to a royal hands, 50.37% will pay nothing, 24.89% will pay as a pair, 7.89% as a straight, 16.16% as a flush, and 0.69% as a straight flush. Here are the exact numbers.

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%.

Good question. In 9/6 jacks or better the probability of a royal flush is 22.65% of that of a straight flush, but a royal pays 16 times more. Overall the straight flush only contributes 0.55% to the return of the game. The straight flush is the Rodney Dangerfield of most forms of video poker, it gets no respect. I can only speculate that game makers wanted a big top prize. Nobody likes to come in second, so perhaps that is why the original game makers didn’t pay the straight flush much by comparison.

9/6 Jacks or Better can still be found at most casinos, although often limited to the high limit room. I suggest VP Free 2 for the latest video poker offerings.

I believe the free Winpoker demo will do that and never expire. The download is available at www.zamzone.com.

I assume that you assume the probability of a royal is 1 in 40,000. Playing 4,000 hands per session the expected number of royals per session is 0.1. A very close appoximation for the probbility of zero royals per session is e^-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.

If you were playing single line it would be easy. $800,000 is 160,000 $5 hands. That is 3.9616 royal cycles. The probability of no royals can be closely approximated as e^-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%.

I knew somebody would eventually write about that. It wasn’t I who showed that machine. The producers didn’t understand I was referring to 9/6 Jacks or Better. Later in the editing room they showed somebody pointing to a 9/6 Double Double Bonus Poker game, which has a return of 98.98%. That is much worse than 9/6 Jacks or Better, at 99.54%. An embarrassing moment for me, much like the many incorrect edits I had no control over, in my old Casino Player articles.

You’re welcome. I ran some random simulations in 9/6 Jacks or Better, to get at the answer to your question. The following table shows the covariance for 2 to 9 lines played, in 9/6 Jacks or Better. The variance would be the same as the base game played.

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.

I feel your pain. 9/6 Jacks or Better is getting harder to find, even at the locals casinos, but they still definitely exist. Some of the MGM/Mirage properties have 9/6 Jacks in their high limit rooms. The Wynn is king of 9/6 Jacks; they have them all over the place. For information about current video poker offerings in Las Vegas, I highly recommend VP Free 2.

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

Playing the optimal 9/6 strategy in this game will result in a return of 0.999796. That is an error rate of only 0.02%, which is not worth learning a new strategy over, in my opinion.

I also play 26 lines at the $1 denomination frequently. The reason is if you get a win of $1,200 or more it necessitates a hand pay, which slows down your game, and obligates you to tip. At 26 lines, a dealt full house in 9/6 jacks, which I happen to know is what he was playing, will pay $5 × 9 × 26 = $1,170. One more line and you would have a hand pay at $1,200. If 26 lines, or $130 a bet, is too small, I’ll go up to 39 lines, where a dealt flush will pay $5 × 6 × 39 = $1,170. The next bend-point is at 59 hands, where a deal straight would be $5 × 4 × 59 = $1,180. However, I feel with 59 hands a three of a kind on the deal turns into a hand pay too often.

For the benefit of other readers, the coefficient of skewness (skew) for any random variable is a measure of which direction has the longer tail. A negative skew means the most likely outcomes are on the high side of the distribution, offset by the extremes tending to be on the low side. A positive skew is the opposite, where the most likely outcomes are the low side, but with the extremes tending to be on the high side. The mean is less than the median with a negative skew, and greater with a positive skew. An exact formula can be found at Wikipedia, or lots of statistics books.

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.

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.

Under your assumptions, you should quit after being up at least one unit, or down 17 units. Using Cramer’s Rule, we can find the the expected number of plays to achieve either marker is 19.227. The probability that the marker achieved is the 17 unit loss is 17.89%. So, the expected refund is 0.1789 × 17 = 3.041076 units. The expected loss of playing 19.227 times on a game with a 0.004561 house edge is 19.227 × 0.004561 = 0.087693 units. So, the expected profit is 3.041076 - 0.004651 = 2.953382 units.

The other video poker player is right. Here is the return table for 9/6 Jacks or Better, breaking down the four of a kinds, assuming optimal strategy.

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

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%.

The following table shows the probability of each kind of royal, according to the number of cards held, given that there was a royal. It shows that 3.4% of royals are from holding one card. The probability of a royal to begin with is 1 in 40,391, so the unconditional probability of a royal holding one card is 1 in 1,186,106.

There are combin(52,5)=2,598,960 possible combinations on the deal. The reason my video poker return tables have almost 20 trillion combinations is you also have to consider what could happen on the draw. Here are the number of combinations according to how many cards the player discards.

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.