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).
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.
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.
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.
|Four of a kind||125|
|Three of a Kind||15|
|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.
|Four of a kind||0%||125||0|
|Three of a Kind||0%||15||0|
|Jacks or Better||0%||5||0|
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.
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.
ducklife on March 15, 2019
EricBurkholz01 on March 10, 2019
HPG ADMIN on March 1, 2013