# Video Poker Strategy

Video poker machines are popular in casinos. They differ from slot machines (as we will see later), but often live side-by-side with them on the floor.

Video poker follows the rules and rankings of traditional five card draw poker, but instead of betting against other players, hands are rewarded based purely on their rank, with the less likely occurring hands scoring higher payouts (For an in-depth look at the breakdown of poker rankings, and their probabilities, check out this article).

After the player makes a wager, the machine deals the player five cards. The player can then hold anywhere from zero to five of these cards, and draw again. The replacement cards are pulled from undealt cards (so there is no chance that the player can reacquire a previously discarded card). If the resulting hand is a winning hand then a payout is made based on the quality of the hand and the bet size.

## Jacks or Better

There are different variants of video poker, but the classic version is called

*Jacks or Better*; this refers to the fact that the machine only pays out on pairs that are formed by Jacks or higher. I’ll be looking exclusively at the Jacks of Better variant in this article as this corresponds most closely with traditional draw poker (The only real difference in the rankings is that, in video poker, a distinction is made between a Royal Flush, and a Straight Flush; A Royal Flush paying out a*much*higher premium. This is important to understand for optimal play). Other common video poker variants are

*Deuces Wild*(in which twos act as wild cards),*Aces and Eights*(Additional bonus payouts for four Aces or Eights, plus a reduced bonus for four Sevens), and*Double Bonus*(This offers increased wins for all four of a kinds with a big payoff for four Aces).## Betting and Paytable

Typically, a player inserts between one and five coins (bets). If playing video poker, it’s foolish not to bet the maximum number of coins. The reason is related to the top payout. Look at the paytable shown below:

If you play just one coin, if you hit a

*Royal Flush*, you are only paid out at 250:1. However, if you play five coins, the same hand will pay out at 800:1. This is a__huge__difference. If you only make the single wager, you are playing into the casinos hand as they are paying out less, and lowering your expected return. For the rest of this article, it’s assumed that you will be making the max bet, but I have normalized the odds back to a bet of one unit to make the math easier to interpret.Below is a payout table for the machine I will be modelling. For reference, I have also included the number of ways these hands can be formed. There are a total of 52

*C*5 = 2,598,960 possible starting hands (the ordering is not important). However, permutations increase rapidly as, for each of these hands, there are 2^{5}= 32 possible ways to play each, and then with a discard of*d*cards there are a further 47*C*d possibilities for the replacements.Rank | Payoff Odds | Number of ways |
---|---|---|

Royal Flush | 800:1 | 4 |

Straight Flush | 50:1 | 36 |

Four of a Kind | 25:1 | 624 |

Full House | 9:1 | 3,744 |

Flush | 6:1 | 5,108 |

Straight | 4:1 | 10,200 |

Three of a Kind | 3:1 | 54,912 |

Two Pair | 2:1 | 123,552 |

Jacks or Better | 1:1 | 337,920 |

Garbage | 0:1 | 2,062,860 |

TOTAL | 2,598,960 |

This is the

*standard*payout table, sometimes called*9/6 poker*or*Full pay poker*in reference to the nine-times odds paid for a Full House, and six-times paid for a Flush. Casinos, however, sometimes adjust these payouts down, for instance to*8/5*, to increase their take. Pay attention!## Skill

Video poker differs from slots in that there is a component of skill. In a slot machine, you insert your money, and luck alone (more specifically, a Random Number Generator), determines the outcome. Nothing you can do will affect the outcome of a slot pull!

In video poker, you make choices. This means you have control of the outcome. If you make the correct choices you can lower the percentage that the casino takes. (Conversely, if you make poor choices, such as throwing away and redrawing a good hand, you can lose money at a much faster rate!)

If played optimally, Jacks or Better video poker (with a 9/6 payout table) has an expected payout of 99.543904%. This is some of the best odds you will find for any casino game.

*If played optimally, Jacks or Better video poker has an expected payout of 99.54%*

However, they key here is that this return is for

*optimal*play. As we will see below, some of the odds and strategy associated with video poker are not obvious, and sometimes counterintuitive. A casino makes better returns on video poker than the theoretical because players don’t play the best way (and also don’t make the maximum wager).The fact that the payout table is displayed, full and clear, to the player is another distinction between video poker and slots. With the correct analysis, and the payout table, you are able to determine the exact expected return from a video poker machine. This is not the case for a slot. Sure, a slot machine has a payout table printed on the side, but you have no idea what the relative frequency of the occurrence of any of these combinations might be.

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## Try it out

Before we delve into some of the mathematics, why not give the game a try?

Below is a simple Jacks or Better poker game I wrote. You start off with 100 credits. How well can you do? To allow you to experiment, I’ve included a cheat matrix at the bottom. This will allow you to select any card, to any position, after the initial deal. Needless to say, you will not find this matrix on a machine in Vegas! (You can also adjust the cards after the draw, to see how the hands would have ranked, but this does not adjust the payout; that gets awarded immediately after the draw).

## Analysis

As mentioned earlier, there are a total of 52

*C*5 = 2,598,960 ways the starting cards can be deait. After this the player makes a choice. The player can hold anywhere from zero to five of the cards. This equates to 2^{5}= 32 possibe holding choices.An easy way to visualize this is to to imagine the cards as bits in a binary number, with

*1*representing a card that will be held, and*0*representing a card that will be redrawn. You can see that all 32 permutations of the hold/drawing combinations can be represented by the numbers*00000-11111*.There are 32 possible ways to play any dealt hand in video poker.

## Expected Value

To work out the optimal play strategy we need to calculate the expected return of each possible 32 hold choices from the cards dealt. We know the payout odds from the payout table, so we evaluate the chance of achieving each of these scoring hands after the draw, and what the expected payout would be for that combination. The superposition (logical or) of all the possible hand outcomes gives us the expected result of that holding pattern. We then select the hold choice that has the largest expected return.

*The optimal play strategy is to select the holding pattern of cards which gives the highest expected value of return after the draw.*

Some of the things we need to be aware of that affect the holding strategies are:

- The payout table is top heavy. The payout for a full-bet Royal Flush is very high. Even though the chance of hitting this might be low, the payout is so high that the expected odds might advise us to go for it.
- It's a superposition of odds, so judegous holding can keep you in the game to potentialy hit against more than one rank after the draw (for instance a high end straight draw may pay out even if you don't make the straight but match one of the top cards to make a Jacks or Better pair). You have more outs with some holds.
- You are playing against the machine, not another player. There is no bigger advantage to having a pair of Aces over a pair of Jacks; they both payout the same. Similarly with three of a kind, or quads.
- Simple pairs below Jacks pay nothing.

Let's look at a few examples. As we will see, sometimes the correct strategy is obvious, sometimes not so obvious …

♦ ♥ ♠ ♣

## Example 1

Imagine your are dealt the following hand, what would you hold?

Well, this is hand is four cards to a Flush. It's also four cards to a Straight. There are also multiple high cards in the hand. Is it better to hold one/more of these? Let's take a look …

Let's look at the potential Straight first. As a traditional poker player, you'd think about holding the A♥ Q♣ J♥ T♥ (holding out for one of the four unseen Kings). There are 47 unseen cards, and there are 4 Kings in this deck, so there is a 4/47 of hitting a King. A Straight pays out at four times the initial wager.

But that's not all. If we don't draw a King, we can still win if we pair the Ace, Queen, or Jack to make Jacks or better. There are nine outs here (The three unseen Aces, the three Queens, and the three Jacks). A Jacks or Better Pair returns the bet of one unit. Any of the other 34 (=47-4-9) cards drawn results in a garbage hand, which does not contribute to the expected return.

The expected result for holding the A♥ Q♣ J♥ T♥ is thus:

As we can see, this is not great (and also quite a bit below 1.0)

However, we can do better. In video poker, a straight, is a straight, is a straight. We're not competing against other players; we don't care if it is a

*Broadway*(a poker colloquial term for the nut straight AKQJT), or any other straight. If, instead of folding the Nine, we fold the Ace, then we are left with Q♣ J♥ T♥ 9♥, which is an open-ended straight draw (We can complete the straight in two ways; either with a King, as before, but now with an Eight). We've doubled our chances of getting straight! We have, however, also reduced our chances of a getting a Jacks or Better pair. Let's see how this affects the expected outcome.This is quite an improvement! But it's still below 1.0, and we can do better. Let's look at the Flush draw:

If we hold A♥ J♥ T♥ 9♥ then, as there are 13 hearts in total, there are nine unseen hearts in the 47 card deck. A Flush also has a higher payout of six times the wager. We also still have the chance of pairing the Ace or Jack (six outs), if we don't make the Flush. The calculation for the expected return for the hold of A♥ J♥ T♥ 9♥ is:

This is a much better strategy, and results in an expected return higher than 1.0 (This does

*not*mean you will win everytime, but if you encountered this situation a larger number of times, and always played this strategy, on average you would end up with approximately 1.27× your wager).The above are three strategies involving discarding just one card. How about discarding more than one? This sounds a little crazy, but let's take a look. What if we hold the J♥ T♥ 9♥ and draw two new cards (The logic behind this is sound, we're holding cards that could cause us to hit both the Straight and/or the Flush, nice!) Let's look at expected result from this strategy.

Holding these three cards opens up the potential for a Straight Flush draw. It's also open ended. We can complete the Straight Flush with either a K♥ Q♥ draw (in any order), a Q♥ 8♥ draw (ditto), or a 8♥ 7♥ draw (ditto). A Straight Flush pays out 50× odds. If we don't hit the Straight Flush, we can still hit an vanilla (unsuited) Straight if we hit the above KQ, Q8, 87 pairs in all the other combinations of suits; a vanilla Straight pays out 4×. There's also still a chance of pairing the Jack, and, as added bonus, since we are drawing

*two*cards, we could also draw a paying Pair of Aces, Kings, or Queens. The love does not stop there! We could draw two Jacks to make Three of a Kind in Jacks, but also two Tens, or two Nines, to makes*'trips'*in those (Three of a Kind pays out 3× odds and, unlike pairs, is agnostic of the denomination).Things are looking promising with all these outs. Let's see how the math works out. It's a little more complex this time.

We'll work with probabilities, initially, instead of expected values, as we have to do some boolean arithmetic when various combinations overlap. First let's look at the chance of drawing a Straight Flush. We can do this in six ways: We can first draw the K♥, followed by the Q♥ (or the other way around), or we can draw Q♥ 8♥ (in two ways), or 8♥ 7♥ (in two ways). Here's the calculation:

Next we'll look at the chance of a Flush. There are nine outstanding hearts from the 47 left in the deck, and we first need to draw one of these, followed by one of the eight remaining from the 46 cards. However, we have to remember that this includes the Straight Flush, so we have to subtract the probability of obtaining that. (We only get paid out on the higher of any matching rank). Here's that calculation:

Next we'll look at the chances of obtaining a Straight. This can be achieved in six ways, similar the the Straight Flush with the correct union of the appropriate of Kings, Queens, Eights, or Sevens, but remember that we discarded the Q♣ from the initial draw, so whilst there are four Kings, Eights, and Sevens, there are only three Queens left in the deck. Finally we need to subtract the chance that we might have gotten a Straight Flush.

There is no chance of drawing to a Full House with this hold, but there is a chance of obtaining Three of a Kind. This can be obtained three way, drawing two additional Jacks, or two additional Tens, or two additional Nines.

Next we need to consider Two Pair. We can draw this by matching two of the three hold cards with the draw. There are nine cards that help us on the first pull (any of the three Jacks, three Tens, or three Nines). After this, there are six remaing cards that will result in Two Pair.

Finally, we need to calculate the chance of obtaining a Pair of Jacks or better. We can do this by pairing the Jack we already hold, or drawning any two Queens (remembering that we already discarded one), any two of the Kings, or any two Aces (again remembering we discarded one). The calculation here is very subtle. There are two ways to pull a single Jack (either on the first card, or the second), but we need to make sure the other card pulled is not another Jack (this would result in a Three of a Kind), and also it cannot be a Ten or a Nine (otherwise we would move into Two Pair scoring). So there are three Jacks in 47 cards, and 38 safe cards (=46-3-3-2) in either of two ways. We can pull a pair of Aces, a pair of Kings, or a pair Queens also.

We now have all the probabilites and can multiply each by the payout and sum to obtain the expected value for holding these three cards.

The result is a, rather disappointing, expected value of approx 0.65 This is not as good as some of the early holding patterns.

Lastly, let's look at holding the three cards A♥ J♥ T♥. This introduces the option a Royal Flush (not present in the holding above), but this is at the expense of the open ended straight draw (in fact, as we are holding the A and the T, the only Straight Flush draw will be a Royal Flush, and the only Straight draw will be an mixed suited Broadway). I'll skip the intermediate probabilites and just shown the expected result formula. It's similar to the calculations above.

This is a good expected return, and just a hair below the expected value of holding four cards to the flush. It shows the awesome weighting of the 800× premium provided by the Royal Flush payout (woe is you if you did not bet max coins). In fact this is so close that if the variant of the games has a progressive jackpot payout for a Royal Flush, then it could flips the strategy over to make holding these three cards have a higher expected value that holding four cards to the flush. If the payout for a Royal Flush was increased to just 808× (equivalent to a progressive jackpot of 4,040

*cf*the standard 4,000), then this would be the correct holding strategy.Without going through all the manual calculations, here is a table showing all possible 32 holding patterns, and the expected outcomes. They are ordered from the top with taking no draws, through drawing 1,2 … 5 cards. The optimal holding strategy is to take the action that results in the highest expected return. It's an important exercise to enumerate them all because, as we have seen, some of the results are not intuitively obvious. This table confirms that the holding strategy we determined of holding four cards to the Flush, is the optimal play if dealt these cards.

Cards Held | Draw | E_{x} | ||||
---|---|---|---|---|---|---|

A♥ | Q♣ | J♥ | T♥ | 9♥ | 0 | 0.000000 |

A♥ | Q♣ | J♥ | T♥ | – | 1 | 0.531915 |

A♥ | Q♣ | J♥ | – | 9♥ | 1 | 0.191489 |

A♥ | Q♣ | – | T♥ | 9♥ | 1 | 0.127660 |

A♥ | – | J♥ | T♥ | 9♥ | 1 | 1.276596 1^{st} |

– | Q♣ | J♥ | T♥ | 9♥ | 1 | 0.808511 3^{rd} |

A♥ | Q♣ | J♥ | – | – | 2 | 0.441258 |

A♥ | Q♣ | – | T♥ | – | 2 | 0.338575 |

A♥ | Q♣ | – | – | 9♥ | 2 | 0.294172 |

A♥ | – | J♥ | T♥ | – | 2 | 1.269195 2^{nd} |

A♥ | – | J♥ | – | 9♥ | 2 | 0.493987 |

A♥ | – | – | T♥ | 9♥ | 2 | 0.391304 |

– | Q♣ | J♥ | T♥ | – | 2 | 0.427382 |

– | Q♣ | J♥ | – | 9♥ | 2 | 0.382979 |

– | Q♣ | – | T♥ | 9♥ | 2 | 0.280296 |

– | – | J♥ | T♥ | 9♥ | 2 | 0.650324 |

A♥ | Q♣ | – | – | – | 3 | 0.464693 |

A♥ | – | J♥ | – | – | 3 | 0.495776 |

A♥ | – | – | T♥ | – | 3 | 0.361640 |

A♥ | – | – | – | 9♥ | 3 | 0.352760 |

– | Q♣ | J♥ | – | – | 3 | 0.482455 |

– | Q♣ | – | T♥ | – | 3 | 0.348319 |

– | Q♣ | – | – | 9♥ | 3 | 0.339439 |

– | – | J♥ | T♥ | – | 3 | 0.391243 |

– | – | J♥ | – | 9♥ | 3 | 0.382362 |

– | – | – | T♥ | 9♥ | 3 | 0.266482 |

A♥ | – | – | – | – | 4 | 0.434614 |

– | Q♣ | – | – | – | 4 | 0.450784 |

– | – | J♥ | – | – | 4 | 0.436722 |

– | – | – | T♥ | – | 4 | 0.272279 |

– | – | – | – | 9♥ | 4 | 0.275822 |

– | – | – | – | – | 5 | 0.308626 |

This table shows that holding all the initial cards (no draw) will result in no payback, and sometimes drawing two, three, or four cards can result in a better expected hand than a poorly selected single draw. Even redrawing the entire hand can be better than some of the other combinations of holds.

From this hand there are only two hold patterns with an expected return higher than 1.0

## Example 2

Our next example is similar to the one above; high cards, and a potential staight, but only three 'painted' hearts, instead of four; decreasing the flush draw. Let's see how thie effects the odds (I'll dispense with showing the intermediate calculations and simple show the top holding strategies).

Here are the top five holding strategies (this time in descending order). Again, the optimal hold is A♥ J♥ T♥ (holding for a potential Royal Flush, as well as a regular Flush, Straight, or High card match). Interestingly, the expected return for this hand is slightly higher than the first example! This is because there is one less heart in the hand and thus increases the chances of drawing a heart. For this deal, it's also the only holding pattern than has an expected value of higher than 1.0

Cards Held | Draw | E_{x} | ||||
---|---|---|---|---|---|---|

A♥ | – | J♥ | T♥ | – | 2 | 1.319149 |

– | Q♣ | J♥ | T♥ | 9♠ | 1 | 0.808511 |

A♥ | Q♣ | J♥ | T♥ | – | 1 | 0.531915 |

A♥ | – | J♥ | – | – | 3 | 0.509097 |

– | Q♣ | J♥ | – | – | 3 | 0.482455 |

Interestingly, if drawing three, the better strategy is to hold A♥ J♥ (because of the potential of getting a Flush, as well as matching Jacks of better), but if they were not suited, then it's better to hold the Q♣ J♥ (as can be hinted at in the row lower), and these cards give the potential for an open-ended Straight draw, whilst the Ace does not).

Another interesting thing (that we'll see in more examples below) is that when there are a couple of high cards in the hand, it's better to hold two of them to potentialy match instead of one. Even though you've one less draw, there are more outs if you hold two potential cards to match against. If you have three high cards to match

*e.g.*AQJxx, depending on their suits, and the other cards, you might potentially hold the Ace, but if not painted, it's better to hold the QJ.## Example 3

In this example, if there are three high cards to the straight, then it's better to hold all three (and if you did want to hold two, then select the two high, connected, cards, the QJ).

Cards Held | Draw | E_{x} | ||||
---|---|---|---|---|---|---|

– | K♣ | Q♠ | J♦ | – | 2 | 0.515264 |

– | – | Q♠ | J♦ | – | 3 | 0.495282 |

– | K♣ | – | J♦ | – | 3 | 0.479494 |

## Example 4

When dealt a low pair, even if this pair is below Jacks (and currently not scoring), it is better to hold the pair (with a chance of making trips, quads, a full house, or two pair), than redraw them all.

Cards Held | Draw | E_{x} | ||||
---|---|---|---|---|---|---|

8♦ | 8♥ | – | – | – | 3 | 0.823682 |

– | – | – | – | – | 5 | 0.364421 |

In the example above, even if the 5♠ were replaced with the A♠, it is still better to holds the 8's than hold the Ace (in fact the Expected return is the same as two pair and above do not have the Jacks or above restriction). If you elected to burn all the cards and draw them all again, instead of holding the 8's, your expected return is less than half of holding them!

## Example 5

This example reinforces the fact that holding the Ace is not optimal (not allowing for an open-ended straight draw), and that it is better to hold two high cards than one.

Cards Held | Draw | E_{x} | ||||
---|---|---|---|---|---|---|

J♦ | Q♠ | – | – | – | 3 | 0.495282 |

– | Q♠ | A♣ | – | – | 3 | 0.467653 |

## Example 6

However, if the Ace and King are painted, this overrides the precidence to hold for the open-ended straight.

Cards Held | Draw | E_{x} | ||||
---|---|---|---|---|---|---|

A♥ | K♥ | – | – | – | 3 | 0.577428 |

– | K♥ | J♣ | – | – | 3 | 0.483441 |

## Example 7

Here's an example showing the awesome weighting effect of the Royal Flush payout. Imagine you've been dealt the following hand. It's a

*pat*hand and is already a Flush; paying out 6× your wager. At first blush, you might be tempted to hold all the cards for the guaranteed win. However, this is not optimal. It's better (in the long run), to fold the small club for the chance to make a Royal Flush. Yes, the chances are low that you'll hit the Ten (1 chance in 47), but the payback on this is over a hundred and thirty times greater than a Flush. Also, even if you don't hit the Royal Flush, there are still other clubs in the deck that could bring you back to a Flush, so this increases the expected return.Cards Held | Draw | E_{x} | ||||
---|---|---|---|---|---|---|

A♣ | K♣ | Q♣ | J♣ | – | 1 | 18.425532 |

A♣ | K♣ | Q♣ | J♣ | 3♣ | 0 | 6.000000 |

Would you have the determination and confidence to burn the three?

## Optimal Hand Calculator

Putting this all this knowledege together, my next idea was to generate an optimal play strategy guide that advised of the correct holding strategy based on any dealt hand. However, as I was working on this I did some research and found out that some smart people have already doene this. Here are two papers on their brilliant work: Video Poker Analysis and Optimal conditional expectation
at the video poker game Jacks or Better.