Video Poker Odds

Introduction

The math behind the video poker odds is not complicated. But given the number of calculations it will take a long time - much too long to do by hand. There are several types of video poker software and simulators on the market that can help you with this.

To find out what the optimal move is to make with a given hand, you need to calculate the expected value for every possible move for the given hand. Calculating the expected return in video poker is like calculating the "implied odds" in regular poker - except the value of the final pot is known (since it is on the payout table).

Process

Starting with a given 5-card hand, list all of the 32 decisions you can make with this hand.
Beginning with the first decision, find the expected value for that decision. This is done by adding together the expected values of all the possible outcomes for that decision. The expected value for each possible outcome is calculated by taking the probability of hitting a particular hand and multiplying it by the payoff. Things get very mathematically intense when you choose to discard more than 1 card.
Repeat the calculations for the other 31 choices for that hand.
Sort all of the 32 choices by the combined expected value. The highest combined expected value is the optimal move for that particular starting hand.

To find the overall odds for the game of video poker, you need to calculate the highest expected value for every single starting hand by calculating the expected value for every possible move with a particular hand. Then, to make it easier, you can group any similar hands together. Given this information, you can create rules to play by. For example, a "made flush" has a higher value than "4 to a straight flush". Then, you will need to note any exceptions to the rules. For example, there may be times where "4 to a flush" will beat "3 to a royal flush", but there may be other times (depending on the particular cards you are dealt) where the opposite is true.

Counting cards

Blackjack players who are not familiar with video poker may wonder if counting cards would be of any value. It won't - for 2 reasons. First, because video poker bases its winnings on the absolute value of a hand (based on a payout table) versus the relative value of a hand (versus a dealer hand), there are no cards to count from other people. The second reason is that the deck is shuffled after every single hand so you can't count your own cards. Some people have said that video poker is equivalent to a blackjack game with zero deck penetration where you are the only player. This is technically incorrect because in a game like that you are can still see the dealer's cards.

Example

It is good just to take a single decision on a single hand and calculate the odds by hand just to see the logic behind the final numbers. There may be times were you may want to do a quick calculation to see the optimal move between only two decisions.

In this example, assume you are playing a game with the following payout schedule for "9/6 Jacks or Better" and assume you are playing 5 credits at a time. Then assume you are dealt the 2, 3, 4, 5 of clubs, plus the 8 of diamonds.

"9/6 Jacks or Better" payout schedule

Hand	5 credits
Royal Flush	4,000
Straight Flush	250
Four of a kind	125
Full House	45
Flush	30
Straight	20
Three of a Kind	15
Two Pair	10
Jacks or Better	5

In this example, we'll find the expected value of the final hand if we discard the 8 of diamonds. Starting at the top of the payout schedule, calculate the probability of hitting each hand.

Royal flush - It isn't possible to make a royal flush.
Straight flush - There are two cards (the Ace and 6 of clubs) that can make a straight flush. The chances of picking one of those two cards out of the 47 cards remaining in the deck are 4.255% (.04255).
Four of a kind - It isn't possible to make four of a kind.
Full House - It isn't possible to make a full house.
Flush - There are 7 (7, 8, 9, 10, J, Q, and K of clubs) cards which can make a flush that were not already included in the calculation for the straight flush. We don't count the A or 6 of clubs because those 2 cards were already included in the calculation for the straight flush.
Straight - There are 6 (the Ace of diamonds, spades, and hearts - and the 6 of diamonds, spades, and hearts) cards that can be used to make a straight that were not already included in the calculation for the straight flush or flush. We don't count the Ace or 6 of clubs because they were used in the calculation of the straight flush.
Three of a Kind - It isn't possible to make three of a kind.
Two Pair - It isn't possible to make two pair.
Jacks or Better - It isn't possible to make a pair of Jacks or better.

Expected value for each potential outcome

Hand	Probability	Payoff	Expected Value
Royal Flush	0%	4,000	0
Straight Flush	0.04255%	250	10.638
Four of a kind	0%	125	0
Full House	0%	45	0
Flush	0.14894%	30	4.468
Straight	0.12766%	20	2.553
Three of a Kind	0%	15	0
Two Pair	0%	10	0
Jacks or Better	0%	5	0
Total			17.660

So, in our example, we add the expected values for a straight flush (10.638), flush (4.468), and a straight (2.553) together to get a toal expected value of 17.660. A couple of other important things . . .

First, it should also be noted that I picked an easy decision where I only discarded a single card. The calculations get exponentially more difficult as you discard more cards.

Second, I picked a starting hand where it was easier to "eyeball" the optimal move, which allows intelligent players to skip doing any needless calculations on any obivously worthless outcomes. Most other hands won't be this easy.

Notes

The calculation that I showed above happens to be the optimal decision for that particular hand but it was only 1 out of 32 decisions that you could have made. In order to know (without having already been told) that it was the optimal move for the hand, you would had to have completed all of the calculations for all of the other decisions so you could see which decision offered the highest expected value.

The table below shows the expected value for every one of the decisions that you could have made regarding the hand and ranks them from the best-to-worst decisions. A red box means that you would discard that card. The first decision on the list is the one we just calculated by hand.

Comparison of expected values for the example hand

Cards					Expected Value
2C	3C	4C	5C	8D	17.6595
2C	3C	4C	5C	8D	2.4098
2C	3C	4C	5C	8D	1.9288
2C	3C	4C	5C	8D	1.9288
2C	3C	4C	5C	8D	1.9288
2C	3C	4C	5C	8D	1.8202
2C	3C	4C	5C	8D	1.6527
2C	3C	4C	5C	8D	1.5404
2C	3C	4C	5C	8D	1.5177
2C	3C	4C	5C	8D	1.5016
2C	3C	4C	5C	8D	1.4854
2C	3C	4C	5C	8D	1.4126
2C	3C	4C	5C	8D	1.3410
2C	3C	4C	5C	8D	1.3410
2C	3C	4C	5C	8D	1.2818
2C	3C	4C	5C	8D	1.2818
2C	3C	4C	5C	8D	1.2818
2C	3C	4C	5C	8D	1.1758
2C	3C	4C	5C	8D	1.0968
2C	3C	4C	5C	8D	1.0376
2C	3C	4C	5C	8D	1.0376
2C	3C	4C	5C	8D	0.7817
2C	3C	4C	5C	8D	0.4857
2C	3C	4C	5C	8D	0.4857
2C	3C	4C	5C	8D	0.4857
2C	3C	4C	5C	8D	0.4857
2C	3C	4C	5C	8D	0.4857
2C	3C	4C	5C	8D	0.0000
2C	3C	4C	5C	8D	0.0000
2C	3C	4C	5C	8D	0.0000
2C	3C	4C	5C	8D	0.0000
2C	3C	4C	5C	8D	0.0000